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Unformatted text preview: SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1 Algebra Review Numbers FRACTIONS Addition and Subtraction i. To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator. ii. To add or subtract fractions with different denominators, find the LCD and write each fraction with this LCD. Then follow the procedure in step i. Definition of Subtraction x - y = x + 1 - y2 Exponents Quotient Rules (continued) Polynomials i. ii. iii. iv. v. (continued) Factoring (continued) Rational Expressions To find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the resulting equation. Rational Expressions (continued) Linear Equations Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an equality without changing the solution. ii. Multiplication: Each side of an equality may be multiplied (or divided) by the same nonzero number without changing the solution. If a Z 0, i. Zero exponent: a0 = 1 ii. Negative exponents: a m -n FOIL Expansion for Multiplying Two Binomials 1 =n a Multiply the first terms. Multiply the outer terms. Multiply the inner terms. Multiply the last terms. Collect like terms. Factoring Trinomials, Leading Term Z x 2 To factor ax2 + bx + c, a Z 1: By Grouping i. Find m and n such that Equations of Lines Two Variables (continued) Intercepts To find the x-intercept, let y = 0. To find the y-intercept, let x = 0. SIMPLIFYING COMPLEX FRACTIONS Method 1 i. Simplify the numerator and denominator separately. ii. Divide by multiplying the simplified numerator by the reciprocal of the simplified denominator. Lowest Terms To write a rational expression in lowest terms: i. Factor the numerator and denominator. ii. Divide out common factors. Slope Suppose (x1, y1) and (x2, y2) are two different points on a line. If x1 Z x2, then the slope is Subtracting Real Numbers i. Change the subtraction symbol to the addition symbol. ii. Change the sign of the number being subtracted. iii. Add using the rules for adding real numbers. iii. Quotient rule: a = am - n an mn = ac and m + n = b. ii. Then ax2 bx c ax2 mx nx c. iii. Group the first two terms and the last two terms. iv. Follow the steps for factoring by grouping. By Trial and Error i. Factor a as pq and c as mn. ii. For each such factorization, form the product 1 px + m21qx + n2 and expand using FOIL. iii. Stop when the expansion matches the original trinomial. iv. Negative to positive: SPECIAL PRODUCTS Square of a Binomial 1 x + y22 = x2 + 2xy + y2 1 x - y22 = x2 - 2xy + y2 Product of the Sum and Difference of Two Terms 1x + y21 x - y2 = x2 - y2 Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial: OPERATIONS ON RATIONAL EXPRESSIONS Multiplying Rational Expressions i. Multiply numerators and multiply denominators. ii. Factor numerators and denominators. iii. Write expression in lowest terms. Solving Linear Equalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. Multiplication Multiply numerators and multiply denominators. Multiplying Real Numbers i. Multiply the absolute value of the two numbers. ii. If the two numbers have the same sign, the product is positve. If the two numbers have different signs, the product is negative. x 1 Definition of Division: = x # , y Z 0 y y Division by 0 is undefined. b a = m , a Z 0, b Z 0 b-n a a -m bm a b = a b , a Z 0, b Z 0 b a Scientific Notation A number written in scientific notation is in the form a * 10 n, where a has one digit in front of the decimal point and that digit is nonzero. To write a number in scientific notation, move the decimal point to follow the first nonzero digit. If the decimal point has been moved n places to the left, the exponent on 10 is n. If the decimal point has been moved n places to the right, the exponent on 10 is –n. -m n Method 2 i. Multiply the numerator and denominator of the complex fraction by the LCD of all the denominators in the complex fraction. ii. Write in lowest terms. m= y 2 - y1 rise = . run x2 - x1 Dividing Rational Expressions i. Multiply the first rational expression by the reciprocal of the second rational expression. ii. Multiply numerators and multiply denominators. iii. Factor numerators and denominators. iv. Write expression in lowest terms. Division Multiply the first fraction by the reciprocal of the second fraction. SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS i. Find the LCD of all denominators in the equation. ii. Multiply each side of the equation by the LCD. iii. Solve the resulting equation. iv. Check that the resulting solutions satisfy the original equation. The slope of a vertical line is undefined. The slope of a horizontal line is 0. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. APPLICATIONS i. Assign a variable to the unknown quantity in the problem. ii. Write an equation involving the unknown. iii. Solve the equation. EQUATIONS OF LINES Slope–intercept form: y = mx + b, where m is the slope, and 10, b2 is the y-intercept. Intercept form: Remainder Theorem If the polynomial P (x) is divided by x – a, then the remainder is equal to P(a). ORDER OF OPERATIONS Simplify within parentheses, brackets, or absolute value bars or above and below fraction bars first, in the following order. i. Apply all exponents. ii. Perform any multiplications or divisions from left to right. iii. Perform any additions or subtractions from left to right. p+q p q = + r r r Dividing a Polynomial by a Polynomial Use long division or synthetic division. Factor Theorem For a polynomial P(x) and number a, if P (a) = 0, then x – a is a factor of P (x). Dividing Real Numbers i. Divide the absolute value of the numbers. ii. If the signs are the same, the answer is positive. If the signs are different, the answer is negative. FORMULAS i. To find the value of one of the variables in a formula, given values for the others, substitute the known values into the formula. ii. To solve a formula for one of the variables, isolate that variable by treating the other variables as constants (numbers) and using the steps for solving equations. Finding the Least Common Denominator (LCD) i. Factor each denominator into prime factors. ii. List each different factor the greatest number of times it appears in any one denominator. iii. Multiply the factors from step ii. Polynomials A polynomial is an algebraic expression made up of a term or a finite sum of terms with real or complex coefficients and whole number exponents. The degree of a term is the sum of the exponents on the variables. The degree of a polynomial is the highest degree amongst all of its terms. A monomial is a polynomial with only one term. A binomial is a polynomial with exactly two terms. A trinomial is a polynomial with exactly three terms. SPECIAL FACTORIZATIONS Difference of Squares x2 - y2 = 1x + y21x - y2 Perfect Square Trinomials x2 + 2xy + y2 = 1x + y22 x2 - 2xy + y2 = 1x - y22 Difference of Cubes x3 - y3 = 1x - y21x2 + xy + y22 Sum of Cubes x3 + y3 = 1x + y21x2 - xy + y22 where 1a, 02 is the x-intercept, and 10, b2 is the y-intercept. Point–slope form: y - y1 = m1x - x12, where m is the slope and 1x1, y12 is any point on the line. Standard form: Ax + By = C Vertical line: x = a Horizontal line: y = b y x + = 1, a b Equations of Lines Two Variables An ordered pair is a solution of an equation if it satisfies the equation. If the value of either variable in an equation is given, the value of the other variable can be found by substitution. Graphing Simple Polynomials i. Determine several points (ordered pairs) satisfying the polynomial equation. ii. Plot the points. iii. Connect the points with a smooth curve. VARIABLES, EXPRESSIONS, AND EQUATIONS An expression containing a variable is evaluated by substituting a given number for the variable. Values for a variable that make an equation true are solutions of the equation. PROPERTIES OF REAL NUMBERS Commutative Properties a+b=b+a ab = ba Associative Properties 1a + b2 + c = a + 1b + c2 1ab2c = a1bc2 Distributive Properties a1b + c2 = ab + ac 1b + c2a = ba + ca Identity Properties a+0=a 0+a=a a#1 = a 1#a = a Inverse Properties a + 1 - a2 = 0 1 - a2 + a = 0 1 1# a# = 1 a = 1 1a Z 02 a a Simplifying Algebraic Expressions When adding or subtracting algebraic expressions, only like terms can be combined. Exponents For any integers m and n, the following rules hold: Writing a Rational Expression with a Specified Denominator i. Factor both denominators. ii. Determine what factors the given denominator must be multiplied by to equal the one given. iii. Multiply the rational expression by that factor divided by itself. GRAPHING LINEAR EQUATIONS To graph a linear equation: i. Find at least two ordered pairs that satisfy the equation. ii. Plot the corresponding points. (An ordered pair (a, b) is plotted by starting at the origin, moving a units along the x-axis and then b units along the y-axis.) iii. Draw a straight line through the points. Factoring Finding the Greatest Common Factor (GCF) i. Include the largest numerical factor of each term. ii. Include each variable that is a factor of every term raised to the smallest exponent that appears in a term. Systems of Linear Equations TWO VARIABLES An ordered pair is a solution of a system if it satisfies all the equations at the same time. REAL NUMBERS AND THE NUMBER LINE a is less than b if a is to the left of b on the number line. The additive inverse of x is –x. The absolute value of x, denoted |x|, is the distance (a positive number) between x and 0 on the number line. ii. 1ab2m = ambm Product Rule am # an = am + n Power Rules i. 1am2n = amn SOLVING QUADRATIC EQUATIONS BY FACTORING Zero-Factor Property If ab = 0, then a = 0 or b = 0. Solving Quadratic Equations i. Write in standard form: OPERATIONS ON POLYNOMIALS Adding Polynomials Add like terms. more➤ Adding or Subtracting Rational Expressions i. Find the LCD. ii. Rewrite each rational expression with the LCD as denominator. iii. If adding, add the numerators to get the numerator of the sum. If subtracting, subtract the second numerator from the first numerator to get the difference. The LCD is the denominator of the sum. iv. Write expression in lowest terms. more➤ Graphing Method i. Graph each equation of the system on the same axes. ii. Find the coordinates of the point of intersection. iii. Verify that the point satisfies all the equations. am am iii. a b = m , b Z 0 b b Subtracting Polynomials Change the sign of the terms in the second polynomial and add to the first polynomial. Factoring by Grouping i. Group the terms. ii. Factor out the greatest common factor in each group. iii. Factor a common binomial factor from the result of step ii. iv. Try various groupings, if necessary. ax2 + bx + c = 0 ii. Factor. iii. Use the zero-factor property to set each factor to zero. iv. Solve each resulting equation to find each solution. OPERATIONS ON REAL NUMBERS Adding Real Numbers To add two numbers with the same sign, add their absolute values. The sum has the same sign as each of the numbers being added. To add two numbers with different signs, subtract their absolute values. The sum has the sign of the number with the larger absolute value. ISBN-13: 978-0-321-39473-6 ISBN-10: 0-321-39473-9 Multiplying Polynomials 90000 Special Graphs x = a is a vertical line through the point 1a, 02. y = b is a horizontal line through the point 1a, b2. The graph of Ax + By = 0 goes through the origin. Find and plot another point that satisfies the equation, and then draw the line through the two points. more➤ Substitution Method i. Solve one equation for either variable. ii. Substitute that variable into the other equation. iii. Solve the equation from step ii. iv. Substitute the result from step iii into the equation from step i to find the remaining value. more➤ i. Multiply each term of the first polynomial by each term of the second polynomial. ii. Collect like terms. more➤ Factoring Trinomials, Leading Term x2 To factor x2 + bx + c, a Z 1: i. Find m and n such that mn = c and m + n = b. ii. Then x2 + bx + c = 1x + m21x + n2. 9 780321 394736 1 2 iii. Verify by using FOIL expansion. 3 more➤ SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1 Algebra Review Numbers FRACTIONS Addition and Subtraction i. To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator. ii. To add or subtract fractions with different denominators, find the LCD and write each fraction with this LCD. Then follow the procedure in step i. Definition of Subtraction x - y = x + 1 - y2 Exponents Quotient Rules (continued) Polynomials i. ii. iii. iv. v. (continued) Factoring (continued) Rational Expressions To find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the resulting equation. Rational Expressions (continued) Linear Equations Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an equality without changing the solution. ii. Multiplication: Each side of an equality may be multiplied (or divided) by the same nonzero number without changing the solution. If a Z 0, i. Zero exponent: a0 = 1 ii. Negative exponents: a m -n FOIL Expansion for Multiplying Two Binomials 1 =n a Multiply the first terms. Multiply the outer terms. Multiply the inner terms. Multiply the last terms. Collect like terms. Factoring Trinomials, Leading Term Z x 2 To factor ax2 + bx + c, a Z 1: By Grouping i. Find m and n such that Equations of Lines Two Variables (continued) Intercepts To find the x-intercept, let y = 0. To find the y-intercept, let x = 0. SIMPLIFYING COMPLEX FRACTIONS Method 1 i. Simplify the numerator and denominator separately. ii. Divide by multiplying the simplified numerator by the reciprocal of the simplified denominator. Lowest Terms To write a rational expression in lowest terms: i. Factor the numerator and denominator. ii. Divide out common factors. Slope Suppose (x1, y1) and (x2, y2) are two different points on a line. If x1 Z x2, then the slope is Subtracting Real Numbers i. Change the subtraction symbol to the addition symbol. ii. Change the sign of the number being subtracted. iii. Add using the rules for adding real numbers. iii. Quotient rule: a = am - n an mn = ac and m + n = b. ii. Then ax2 bx c ax2 mx nx c. iii. Group the first two terms and the last two terms. iv. Follow the steps for factoring by grouping. By Trial and Error i. Factor a as pq and c as mn. ii. For each such factorization, form the product 1 px + m21qx + n2 and expand using FOIL. iii. Stop when the expansion matches the original trinomial. iv. Negative to positive: SPECIAL PRODUCTS Square of a Binomial 1 x + y22 = x2 + 2xy + y2 1 x - y22 = x2 - 2xy + y2 Product of the Sum and Difference of Two Terms 1x + y21 x - y2 = x2 - y2 Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial: OPERATIONS ON RATIONAL EXPRESSIONS Multiplying Rational Expressions i. Multiply numerators and multiply denominators. ii. Factor numerators and denominators. iii. Write expression in lowest terms. Solving Linear Equalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. Multiplication Multiply numerators and multiply denominators. Multiplying Real Numbers i. Multiply the absolute value of the two numbers. ii. If the two numbers have the same sign, the product is positve. If the two numbers have different signs, the product is negative. x 1 Definition of Division: = x # , y Z 0 y y Division by 0 is undefined. b a = m , a Z 0, b Z 0 b-n a a -m bm a b = a b , a Z 0, b Z 0 b a Scientific Notation A number written in scientific notation is in the form a * 10 n, where a has one digit in front of the decimal point and that digit is nonzero. To write a number in scientific notation, move the decimal point to follow the first nonzero digit. If the decimal point has been moved n places to the left, the exponent on 10 is n. If the decimal point has been moved n places to the right, the exponent on 10 is –n. -m n Method 2 i. Multiply the numerator and denominator of the complex fraction by the LCD of all the denominators in the complex fraction. ii. Write in lowest terms. m= y 2 - y1 rise = . run x2 - x1 Dividing Rational Expressions i. Multiply the first rational expression by the reciprocal of the second rational expression. ii. Multiply numerators and multiply denominators. iii. Factor numerators and denominators. iv. Write expression in lowest terms. Division Multiply the first fraction by the reciprocal of the second fraction. SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS i. Find the LCD of all denominators in the equation. ii. Multiply each side of the equation by the LCD. iii. Solve the resulting equation. iv. Check that the resulting solutions satisfy the original equation. The slope of a vertical line is undefined. The slope of a horizontal line is 0. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. APPLICATIONS i. Assign a variable to the unknown quantity in the problem. ii. Write an equation involving the unknown. iii. Solve the equation. EQUATIONS OF LINES Slope–intercept form: y = mx + b, where m is the slope, and 10, b2 is the y-intercept. Intercept form: Remainder Theorem If the polynomial P (x) is divided by x – a, then the remainder is equal to P(a). ORDER OF OPERATIONS Simplify within parentheses, brackets, or absolute value bars or above and below fraction bars first, in the following order. i. Apply all exponents. ii. Perform any multiplications or divisions from left to right. iii. Perform any additions or subtractions from left to right. p+q p q = + r r r Dividing a Polynomial by a Polynomial Use long division or synthetic division. Factor Theorem For a polynomial P(x) and number a, if P (a) = 0, then x – a is a factor of P (x). Dividing Real Numbers i. Divide the absolute value of the numbers. ii. If the signs are the same, the answer is positive. If the signs are different, the answer is negative. FORMULAS i. To find the value of one of the variables in a formula, given values for the others, substitute the known values into the formula. ii. To solve a formula for one of the variables, isolate that variable by treating the other variables as constants (numbers) and using the steps for solving equations. Finding the Least Common Denominator (LCD) i. Factor each denominator into prime factors. ii. List each different factor the greatest number of times it appears in any one denominator. iii. Multiply the factors from step ii. Polynomials A polynomial is an algebraic expression made up of a term or a finite sum of terms with real or complex coefficients and whole number exponents. The degree of a term is the sum of the exponents on the variables. The degree of a polynomial is the highest degree amongst all of its terms. A monomial is a polynomial with only one term. A binomial is a polynomial with exactly two terms. A trinomial is a polynomial with exactly three terms. SPECIAL FACTORIZATIONS Difference of Squares x2 - y2 = 1x + y21x - y2 Perfect Square Trinomials x2 + 2xy + y2 = 1x + y22 x2 - 2xy + y2 = 1x - y22 Difference of Cubes x3 - y3 = 1x - y21x2 + xy + y22 Sum of Cubes x3 + y3 = 1x + y21x2 - xy + y22 where 1a, 02 is the x-intercept, and 10, b2 is the y-intercept. Point–slope form: y - y1 = m1x - x12, where m is the slope and 1x1, y12 is any point on the line. Standard form: Ax + By = C Vertical line: x = a Horizontal line: y = b y x + = 1, a b Equations of Lines Two Variables An ordered pair is a solution of an equation if it satisfies the equation. If the value of either variable in an equation is given, the value of the other variable can be found by substitution. Graphing Simple Polynomials i. Determine several points (ordered pairs) satisfying the polynomial equation. ii. Plot the points. iii. Connect the points with a smooth curve. VARIABLES, EXPRESSIONS, AND EQUATIONS An expression containing a variable is evaluated by substituting a given number for the variable. Values for a variable that make an equation true are solutions of the equation. PROPERTIES OF REAL NUMBERS Commutative Properties a+b=b+a ab = ba Associative Properties 1a + b2 + c = a + 1b + c2 1ab2c = a1bc2 Distributive Properties a1b + c2 = ab + ac 1b + c2a = ba + ca Identity Properties a+0=a 0+a=a a#1 = a 1#a = a Inverse Properties a + 1 - a2 = 0 1 - a2 + a = 0 1 1# a# = 1 a = 1 1a Z 02 a a Simplifying Algebraic Expressions When adding or subtracting algebraic expressions, only like terms can be combined. Exponents For any integers m and n, the following rules hold: Writing a Rational Expression with a Specified Denominator i. Factor both denominators. ii. Determine what factors the given denominator must be multiplied by to equal the one given. iii. Multiply the rational expression by that factor divided by itself. GRAPHING LINEAR EQUATIONS To graph a linear equation: i. Find at least two ordered pairs that satisfy the equation. ii. Plot the corresponding points. (An ordered pair (a, b) is plotted by starting at the origin, moving a units along the x-axis and then b units along the y-axis.) iii. Draw a straight line through the points. Factoring Finding the Greatest Common Factor (GCF) i. Include the largest numerical factor of each term. ii. Include each variable that is a factor of every term raised to the smallest exponent that appears in a term. Systems of Linear Equations TWO VARIABLES An ordered pair is a solution of a system if it satisfies all the equations at the same time. REAL NUMBERS AND THE NUMBER LINE a is less than b if a is to the left of b on the number line. The additive inverse of x is –x. The absolute value of x, denoted |x|, is the distance (a positive number) between x and 0 on the number line. ii. 1ab2m = ambm Product Rule am # an = am + n Power Rules i. 1am2n = amn SOLVING QUADRATIC EQUATIONS BY FACTORING Zero-Factor Property If ab = 0, then a = 0 or b = 0. Solving Quadratic Equations i. Write in standard form: OPERATIONS ON POLYNOMIALS Adding Polynomials Add like terms. more➤ Adding or Subtracting Rational Expressions i. Find the LCD. ii. Rewrite each rational expression with the LCD as denominator. iii. If adding, add the numerators to get the numerator of the sum. If subtracting, subtract the second numerator from the first numerator to get the difference. The LCD is the denominator of the sum. iv. Write expression in lowest terms. more➤ Graphing Method i. Graph each equation of the system on the same axes. ii. Find the coordinates of the point of intersection. iii. Verify that the point satisfies all the equations. am am iii. a b = m , b Z 0 b b Subtracting Polynomials Change the sign of the terms in the second polynomial and add to the first polynomial. Factoring by Grouping i. Group the terms. ii. Factor out the greatest common factor in each group. iii. Factor a common binomial factor from the result of step ii. iv. Try various groupings, if necessary. ax2 + bx + c = 0 ii. Factor. iii. Use the zero-factor property to set each factor to zero. iv. Solve each resulting equation to find each solution. OPERATIONS ON REAL NUMBERS Adding Real Numbers To add two numbers with the same sign, add their absolute values. The sum has the same sign as each of the numbers being added. To add two numbers with different signs, subtract their absolute values. The sum has the sign of the number with the larger absolute value. ISBN-13: 978-0-321-39473-6 ISBN-10: 0-321-39473-9 Multiplying Polynomials 90000 Special Graphs x = a is a vertical line through the point 1a, 02. y = b is a horizontal line through the point 1a, b2. The graph of Ax + By = 0 goes through the origin. Find and plot another point that satisfies the equation, and then draw the line through the two points. more➤ Substitution Method i. Solve one equation for either variable. ii. Substitute that variable into the other equation. iii. Solve the equation from step ii. iv. Substitute the result from step iii into the equation from step i to find the remaining value. more➤ i. Multiply each term of the first polynomial by each term of the second polynomial. ii. Collect like terms. more➤ Factoring Trinomials, Leading Term x2 To factor x2 + bx + c, a Z 1: i. Find m and n such that mn = c and m + n = b. ii. Then x2 + bx + c = 1x + m21x + n2. 9 780321 394736 1 2 iii. Verify by using FOIL expansion. 3 more➤ SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1 Algebra Review Numbers FRACTIONS Addition and Subtraction i. To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator. ii. To add or subtract fractions with different denominators, find the LCD and write each fraction with this LCD. Then follow the procedure in step i. Definition of Subtraction x - y = x + 1 - y2 Exponents Quotient Rules (continued) Polynomials i. ii. iii. iv. v. (continued) Factoring (continued) Rational Expressions To find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the resulting equation. Rational Expressions (continued) Linear Equations Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an equality without changing the solution. ii. Multiplication: Each side of an equality may be multiplied (or divided) by the same nonzero number without changing the solution. If a Z 0, i. Zero exponent: a0 = 1 ii. Negative exponents: a m -n FOIL Expansion for Multiplying Two Binomials 1 =n a Multiply the first terms. Multiply the outer terms. Multiply the inner terms. Multiply the last terms. Collect like terms. Factoring Trinomials, Leading Term Z x 2 To factor ax2 + bx + c, a Z 1: By Grouping i. Find m and n such that Equations of Lines Two Variables (continued) Intercepts To find the x-intercept, let y = 0. To find the y-intercept, let x = 0. SIMPLIFYING COMPLEX FRACTIONS Method 1 i. Simplify the numerator and denominator separately. ii. Divide by multiplying the simplified numerator by the reciprocal of the simplified denominator. Lowest Terms To write a rational expression in lowest terms: i. Factor the numerator and denominator. ii. Divide out common factors. Slope Suppose (x1, y1) and (x2, y2) are two different points on a line. If x1 Z x2, then the slope is Subtracting Real Numbers i. Change the subtraction symbol to the addition symbol. ii. Change the sign of the number being subtracted. iii. Add using the rules for adding real numbers. iii. Quotient rule: a = am - n an mn = ac and m + n = b. ii. Then ax2 bx c ax2 mx nx c. iii. Group the first two terms and the last two terms. iv. Follow the steps for factoring by grouping. By Trial and Error i. Factor a as pq and c as mn. ii. For each such factorization, form the product 1 px + m21qx + n2 and expand using FOIL. iii. Stop when the expansion matches the original trinomial. iv. Negative to positive: SPECIAL PRODUCTS Square of a Binomial 1 x + y22 = x2 + 2xy + y2 1 x - y22 = x2 - 2xy + y2 Product of the Sum and Difference of Two Terms 1x + y21 x - y2 = x2 - y2 Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial: OPERATIONS ON RATIONAL EXPRESSIONS Multiplying Rational Expressions i. Multiply numerators and multiply denominators. ii. Factor numerators and denominators. iii. Write expression in lowest terms. Solving Linear Equalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. Multiplication Multiply numerators and multiply denominators. Multiplying Real Numbers i. Multiply the absolute value of the two numbers. ii. If the two numbers have the same sign, the product is positve. If the two numbers have different signs, the product is negative. x 1 Definition of Division: = x # , y Z 0 y y Division by 0 is undefined. b a = m , a Z 0, b Z 0 b-n a a -m bm a b = a b , a Z 0, b Z 0 b a Scientific Notation A number written in scientific notation is in the form a * 10 n, where a has one digit in front of the decimal point and that digit is nonzero. To write a number in scientific notation, move the decimal point to follow the first nonzero digit. If the decimal point has been moved n places to the left, the exponent on 10 is n. If the decimal point has been moved n places to the right, the exponent on 10 is –n. -m n Method 2 i. Multiply the numerator and denominator of the complex fraction by the LCD of all the denominators in the complex fraction. ii. Write in lowest terms. m= y 2 - y1 rise = . run x2 - x1 Dividing Rational Expressions i. Multiply the first rational expression by the reciprocal of the second rational expression. ii. Multiply numerators and multiply denominators. iii. Factor numerators and denominators. iv. Write expression in lowest terms. Division Multiply the first fraction by the reciprocal of the second fraction. SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS i. Find the LCD of all denominators in the equation. ii. Multiply each side of the equation by the LCD. iii. Solve the resulting equation. iv. Check that the resulting solutions satisfy the original equation. The slope of a vertical line is undefined. The slope of a horizontal line is 0. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. APPLICATIONS i. Assign a variable to the unknown quantity in the problem. ii. Write an equation involving the unknown. iii. Solve the equation. EQUATIONS OF LINES Slope–intercept form: y = mx + b, where m is the slope, and 10, b2 is the y-intercept. Intercept form: Remainder Theorem If the polynomial P (x) is divided by x – a, then the remainder is equal to P(a). ORDER OF OPERATIONS Simplify within parentheses, brackets, or absolute value bars or above and below fraction bars first, in the following order. i. Apply all exponents. ii. Perform any multiplications or divisions from left to right. iii. Perform any additions or subtractions from left to right. p+q p q = + r r r Dividing a Polynomial by a Polynomial Use long division or synthetic division. Factor Theorem For a polynomial P(x) and number a, if P (a) = 0, then x – a is a factor of P (x). Dividing Real Numbers i. Divide the absolute value of the numbers. ii. If the signs are the same, the answer is positive. If the signs are different, the answer is negative. FORMULAS i. To find the value of one of the variables in a formula, given values for the others, substitute the known values into the formula. ii. To solve a formula for one of the variables, isolate that variable by treating the other variables as constants (numbers) and using the steps for solving equations. Finding the Least Common Denominator (LCD) i. Factor each denominator into prime factors. ii. List each different factor the greatest number of times it appears in any one denominator. iii. Multiply the factors from step ii. Polynomials A polynomial is an algebraic expression made up of a term or a finite sum of terms with real or complex coefficients and whole number exponents. The degree of a term is the sum of the exponents on the variables. The degree of a polynomial is the highest degree amongst all of its terms. A monomial is a polynomial with only one term. A binomial is a polynomial with exactly two terms. A trinomial is a polynomial with exactly three terms. SPECIAL FACTORIZATIONS Difference of Squares x2 - y2 = 1x + y21x - y2 Perfect Square Trinomials x2 + 2xy + y2 = 1x + y22 x2 - 2xy + y2 = 1x - y22 Difference of Cubes x3 - y3 = 1x - y21x2 + xy + y22 Sum of Cubes x3 + y3 = 1x + y21x2 - xy + y22 where 1a, 02 is the x-intercept, and 10, b2 is the y-intercept. Point–slope form: y - y1 = m1x - x12, where m is the slope and 1x1, y12 is any point on the line. Standard form: Ax + By = C Vertical line: x = a Horizontal line: y = b y x + = 1, a b Equations of Lines Two Variables An ordered pair is a solution of an equation if it satisfies the equation. If the value of either variable in an equation is given, the value of the other variable can be found by substitution. Graphing Simple Polynomials i. Determine several points (ordered pairs) satisfying the polynomial equation. ii. Plot the points. iii. Connect the points with a smooth curve. VARIABLES, EXPRESSIONS, AND EQUATIONS An expression containing a variable is evaluated by substituting a given number for the variable. Values for a variable that make an equation true are solutions of the equation. PROPERTIES OF REAL NUMBERS Commutative Properties a+b=b+a ab = ba Associative Properties 1a + b2 + c = a + 1b + c2 1ab2c = a1bc2 Distributive Properties a1b + c2 = ab + ac 1b + c2a = ba + ca Identity Properties a+0=a 0+a=a a#1 = a 1#a = a Inverse Properties a + 1 - a2 = 0 1 - a2 + a = 0 1 1# a# = 1 a = 1 1a Z 02 a a Simplifying Algebraic Expressions When adding or subtracting algebraic expressions, only like terms can be combined. Exponents For any integers m and n, the following rules hold: Writing a Rational Expression with a Specified Denominator i. Factor both denominators. ii. Determine what factors the given denominator must be multiplied by to equal the one given. iii. Multiply the rational expression by that factor divided by itself. GRAPHING LINEAR EQUATIONS To graph a linear equation: i. Find at least two ordered pairs that satisfy the equation. ii. Plot the corresponding points. (An ordered pair (a, b) is plotted by starting at the origin, moving a units along the x-axis and then b units along the y-axis.) iii. Draw a straight line through the points. Factoring Finding the Greatest Common Factor (GCF) i. Include the largest numerical factor of each term. ii. Include each variable that is a factor of every term raised to the smallest exponent that appears in a term. Systems of Linear Equations TWO VARIABLES An ordered pair is a solution of a system if it satisfies all the equations at the same time. REAL NUMBERS AND THE NUMBER LINE a is less than b if a is to the left of b on the number line. The additive inverse of x is –x. The absolute value of x, denoted |x|, is the distance (a positive number) between x and 0 on the number line. ii. 1ab2m = ambm Product Rule am # an = am + n Power Rules i. 1am2n = amn SOLVING QUADRATIC EQUATIONS BY FACTORING Zero-Factor Property If ab = 0, then a = 0 or b = 0. Solving Quadratic Equations i. Write in standard form: OPERATIONS ON POLYNOMIALS Adding Polynomials Add like terms. more➤ Adding or Subtracting Rational Expressions i. Find the LCD. ii. Rewrite each rational expression with the LCD as denominator. iii. If adding, add the numerators to get the numerator of the sum. If subtracting, subtract the second numerator from the first numerator to get the difference. The LCD is the denominator of the sum. iv. Write expression in lowest terms. more➤ Graphing Method i. Graph each equation of the system on the same axes. ii. Find the coordinates of the point of intersection. iii. Verify that the point satisfies all the equations. am am iii. a b = m , b Z 0 b b Subtracting Polynomials Change the sign of the terms in the second polynomial and add to the first polynomial. Factoring by Grouping i. Group the terms. ii. Factor out the greatest common factor in each group. iii. Factor a common binomial factor from the result of step ii. iv. Try various groupings, if necessary. ax2 + bx + c = 0 ii. Factor. iii. Use the zero-factor property to set each factor to zero. iv. Solve each resulting equation to find each solution. OPERATIONS ON REAL NUMBERS Adding Real Numbers To add two numbers with the same sign, add their absolute values. The sum has the same sign as each of the numbers being added. To add two numbers with different signs, subtract their absolute values. The sum has the sign of the number with the larger absolute value. ISBN-13: 978-0-321-39473-6 ISBN-10: 0-321-39473-9 Multiplying Polynomials 90000 Special Graphs x = a is a vertical line through the point 1a, 02. y = b is a horizontal line through the point 1a, b2. The graph of Ax + By = 0 goes through the origin. Find and plot another point that satisfies the equation, and then draw the line through the two points. more➤ Substitution Method i. Solve one equation for either variable. ii. Substitute that variable into the other equation. iii. Solve the equation from step ii. iv. Substitute the result from step iii into the equation from step i to find the remaining value. more➤ i. Multiply each term of the first polynomial by each term of the second polynomial. ii. Collect like terms. more➤ Factoring Trinomials, Leading Term x2 To factor x2 + bx + c, a Z 1: i. Find m and n such that mn = c and m + n = b. ii. Then x2 + bx + c = 1x + m21x + n2. 9 780321 394736 1 2 iii. Verify by using FOIL expansion. 3 more➤ SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 2 Algebra Review Systems of Linear Equations (continued) Elimination Method i. Write the equations in standard form: Inequalities and Absolute Value: One Variable Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an inequality without changing the solution. ii. Multiplication by positive numbers: Each side of an inequality may be multiplied (or divided) by the same positive number without changing the solution. iii. Multiplication by negative numbers: If each side of an inequality is multiplied (or divided) by the same negative number, the direction of the inequality symbol is reversed. Inequalities and Absolute Value: One Variable (continued) Graphing a Linear Inequality i. If the inequality sign is replaced by an equals sign, the resulting line is the equation of the boundary. ii. Draw the graph of the boundary line, making the line solid if the inequality involves … or Ú or dashed if the inequality involves < or >. iii. Choose any point not on the line as a test point and substitute its coordinates into the inequality. iv. If the test point satisfies the inequality, shade the region that includes the test point; otherwise, shade the region that does not include the test point. Roots and Radicals n n Ax + By = C. ii. Multiply one or both equations by appropriate numbers so that the sum of the coefficient of one variable is 0. iii. Add the equations to eliminate one of the variables. iv. Solve the equation that results from step iii. v. Substitute the solution from step iv into either of the original equations to find the value of the remaining variable. Notes: If the result of step iii is a false statement, the graphs are parallel lines and there is no solution. If the result of step iii is a true statement, such as 0 = 0, the graphs are the same line, and the solution is every ordered pair on either line (of which there are infinitely many). 2an = a if n is odd. Rational Exponents n n a1>n: If 1 a is real, then a1>n = 1 a. 0 n n 2a is the principal or positive nth root of a. - 2 a is the negative nth root of a. 2a = |a| if n is even. n 2a = b means bn = a. Roots and Radicals The imaginary unit is i = 2 - 1, so i 2 = - 1. Radical Expressions and Graphs (continued) Quadratic Equations, Inequalities, and Functions (continued) Discriminant Number and Type of Solution Two real solutions One real solution Two complex solutions Inverse, Exponential, and Logarithmic Functions (continued) For a 7 0, a Z 1, f1x2 = ax defines the exponential function with base a. Properties of the graph of f1x2 = ax: i. Contains the point (0, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The x-axis is an asymptote. iv. Domain: ( - q , q ); Range: (0, q ) Conic Sections and Nonlinear Systems (continued) Sequences and Series A sequence is a list of terms t1, t2, t3, … (finite or infinite) whose general (nth) term is denoted tn. A series is the sum of the terms in a sequence. For b 7 0, 2 - b = i 2b. To multiply rad- COMPLEX NUMBERS Exponential Functions each factor to the form i 2b. A complex number has the form a + bi, where a and b are real numbers. ELLIPSE Equation of an Ellipse (Standard Position, Major Axis along x-axis) y x + 2 = 1, a 7 b 7 0 a2 b is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1a, 02 and 1 - a, 02 and y-intercepts are 10, b2 10, - b2. Foci are 1c, 02 and 1 - c, 02, 2 2 where c = 2a - b . 2 2 icals with negative radicands, first change b2 - 4ac 7 0 b2 - 4ac = 0 b2 - 4ac 6 0 ARITHMETIC SEQUENCES An arithmetic sequence is a sequence in which the difference between successive terms is a constant. Let a1 be the first term, an be the n th term, and d be the common difference. Common difference: d = an+1 – an nth term: an = a1 + 1n - 12d QUADRATIC FUNCTIONS Standard Form f1x2 = ax2 + bx + c, for a, b, c real, a Z 0. The graph is a parabola, opening up if a : If m and n are positive integers with m/n in lowest terms and a1>n is real, then If a1>n is not real, then am>n is not real. m>n OPERATIONS ON COMPLEX NUMBERS Adding and Subtracting Complex Numbers Add (or subtract) the real parts and add (or subtract) the imaginary parts. Solving Linear Inequalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. (Reverse the inequality symbol when multiplying or dividing by a negative number.) Functions Function Notation A function is a set of ordered pairs (x, y) such that for each first component x, there is one and only one second component y. The set of first components is called the domain, and the set of second components is called the range. y = f(x) defines y as a function of x. To write an equation that defines y as a function of x in function notation, solve the equation for y and replace y by f (x). To evaluate a function written in function notation for a given value of x, substitute the value wherever x appears. Product Rule: If 1a and 1b are real and n is a natural number, then am>n = 1a1>n2m. a 7 0, down if a 6 0. The vertex is b a -a , 4ac 42 a b2 Logarithmic Functions The logarithmic function is the inverse of the exponential function: y = loga x means x = ay. For a 7 0, a Z 1, g1x2 = loga x defines the logarithmic function with base a. Properties of the graph of g1x2 = loga x: i. Contains the points (1, 0) and (a, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The y-axis is an asymptote. iv. Domain: (0, q ); Range: ( - q , q ) SIMPLIFYING RADICAL EXPRESSIONS Quotient Rule: If 1a and 1b are real and n is a natural number, then n n n n n Solving Compound Inequalities i. Solve each inequality in the compound, inequality individually. ii. If the inequalities are joined with and, then the solution set is the intersection of the two individual solution sets. iii. If the inequalities are joined with or, then the solution set is the union of the two individual solution sets. 1a # 1b = 1ab. a 1a =n. Ab 1b n n n n b. -b 2a . Sum of the first n terms: Multiplying Complex Numbers Multiply using FOIL expansion and using i 2 = - 1 to reduce the result. The axis of symmetry is x = Equation of an Ellipse (Standard Position, Major Axis along y-axis) y2 a 2 Sn = Dividing Complex Numbers Multiply the numerator and the denominator by the conjugate of the denominator. THREE VARIABLES i. Use the elimination method to eliminate any variable from any two of the original equations. ii. Eliminate the same variable from any other two equations. iii. Steps i and ii produce a system of two equations in two variables. Use the elimination method for two-variable systems to solve for the two variables. iv. Substitute the values from step iii into any of the original equations to find the value of the remaining variable. Vertex Form f1x2 = a1x - h22 + k. The vertex is 1h, k2. The axis of symmetry is x = h. Horizontal Parabola The graph of x = ay2 + by + c, is a horizontal parabola, opening to the right if + x2 = 1, a 7 b 7 0 b2 n n 1a1 + an2 = 32a1 + 1n - 12d4 2 2 GEOMETRIC SEQUENCES A geometric sequence is a sequence in which the ratio of successive terms is a constant. Let t1 be the first term, t n be the nth term, and r be the common ratio. Common ratio: r = OPERATIONS ON RADICAL EXPRESSIONS Adding and Subtracting: Only radical expressions with the same index and the same radicand can be combined. Multiplying: Multiply binomial radical expressions by using FOIL expansion. Dividing: Rationalize the denominator by multiplying both the numerator and denominator by the same expression. If the denominator involves the sum of an integer and a square root, the expression used will be chosen to create a difference of squares. Quadratic Equations, Inequalities, and Functions If a is a complex number, then the solutions to x2 = a are x = 1a and x = - 1a. a 7 0, to the left if a 6 0. Note that this is not the graph of a function. is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1b, 02 and 1 - b, 02 and y-intercepts are 10, a2 10, - a2. Foci are 10, c2 and 10, - c2, where c = 2a2 - b2. tn + 1 tn SOLVING QUADRATIC EQUATIONS Square Root Property QUADRATIC INEQUALITIES Solving Quadratic (or HigherDegree Polynomial) Inequalities i. Replace the inequality sign by an equality sign and find the real-valued solutions to the equation. ii. Use the solutions from step i to divide the real number line into intervals. iii. Substitute a test number from each interval into the original inequality to determine the intervals that belong to the solution set. iv. Consider the endpoints separately. Solving Absolute Value Equations and Inequalities Suppose k is positive. To solve ƒ ax + b ƒ = k, solve the compound equation ax + b = k or ax + b = - k. To solve ƒ ax + b ƒ 7 k, solve the compound inequality ax + b 7 k or ax + b 6 - k. To solve ƒ ax + b ƒ 6 k, solve the compound inequality - k 6 ax + b 6 k. To solve an absolute value equation of the form ƒ ax + b ƒ = ƒ cx + d ƒ , solve the compound equation ax + b = cx + d or Variation If there exists some real number (constant) k such that: y = kx n, then y varies directly as xn. APPLICATIONS i. Assign variables to the unknown quantities in the problem. ii. Write a system of equations that relates the unknowns. iii. Solve the system. Solving Quadratic Equations by Completing the Square To solve ax2 + bx + c = 0, a Z 0: i. If a Z 1, divide each side by a. ii. Write the equation with the variable terms on one side of the equals sign and the constant on the other. iii. Take half the coefficient of x and square it. Add the square to each side of the equation. iv. Factor the perfect square trinomial and write it as the square of a binomial. Combine the constants on the other side. v. Use the square root property to determine the solutions. MATRIX ROW OPERATIONS i. Any two rows of the matrix may be interchanged. ii. All the elements in any row may be multiplied by any nonzero real number. iii. Any row may be modified by adding to the elements of the row the product of a real number and the elements of another row. A system of equations can be represented by a matrix and solved by matrix methods. Write an augmented matrix and use row operations to reduce the matrix to row echelon form. k , then y varies inversely as xn. xn y = kxz, then y varies jointly as x and z. Operations on Functions If f(x) and g(x) are functions, then the y= following functions are derived from f and g: Solving Equations Involving Radicals i. Isolate one radical on one side of the equation. ii. Raise both sides of the equation to a power that equals the index of the radical. iii. Solve the resulting equation; if it still contains a radical, repeat steps i and ii. iv. The resulting solutions are only candidates. Check which ones satisfy the original equation. Candidates that do not check are extraneous (not part of the solution set). more➤ 1f - g21x2 = f1x2 - g1x2 f 1x2 f a b (x) = , g1x2 Z 0 g g1x2 Composition of f and g: 1f + g21x2 = f1x2 + g1x2 1fg21x2 = f1x2 # g1x2 Inverse, Exponential, and Logarithmic Functions Inverse Functions If any horizontal line intersects the graph of a function in, at most, one point, then the function is one to one and has an inverse. If y = f (x) is one to one, then the equation that defines the inverse function f –1 is found by interchanging x and y, solving for y, and replacing y with f –1(x). The graph of f –1 is the mirror image of the graph of f with respect to the line y = x . more➤ Logarithm Rules Product rule: log a xy = log a x + log a y x Quotient rule: log a y = log a x - log a y r Power rule: log a x = r log a x Special properties: a log a x = x, log a ax = x Change-of-base rule: For a 7 0, a Z 1, logb x b 7 0, b Z 1, x 7 0, log a x = . logb a Exponential, Logarithmic Equations Suppose b 7 0, b Z 1. i. If bx = by, then x = y. ii. If x 7 0, y 7 0, then log b x = log b y is equivalent to x = y. iii. If log b x = y, then by = x. HYPERBOLA Equation of a Hyperbola (Standard Position, Opening Left and Right) y2 x2 - 2=1 2 a b 1a, 02 and 1 - a, 02. Foci are 1c, 02 and 1 - c, 02, where c = 2a2 + b2. b Asymptotes are y = ; a x. Equation of a Hyperbola (Standard Position, Opening Up and Down) y2 a 2 nth term: tn = t1r n - 1 Sum of the first n terms: Sn = t11r n - 12 r-1 ,r Z 1 is the equation of a hyperbola centered at Sum of the terms of an infinite geometric sequence with |r| < 1: S = the origin, whose x-intercepts (vertices) are t1 1-r The Binomial Theorem Factorials For any positive integer n, and - x2 b2 =1 n! = n1n - 121n - 22 Á 132122112 0! = 1. Binomial Coefficient Conic Sections and Nonlinear Systems CIRCLE Equation of a Circle: Center-Radius 1x - h22 + 1y - k22 = r2 Equation of a Circle: General x2 + y2 + ax + by + c = 0 is the equation of a hyperbola centered at the where c = 2a2 + b2. Asymptotes are a y = ; b x. origin, whose y-intercepts (vertices) are 10, a2 and 10, - a2. Foci are 10, c2 and 10, - c2, ax + b = - 1cx + d2. more➤ Quadratic Formula The solutions of ax2 + bx + c = 0, a Z 0 are given by x= -b 1f g21x2 = f 3g1x24 2b2 - 4ac . 2a For any nonnegative integers n and r, with n n! . r … n, a b = nCp = r r!1n - r2! The binomial expansion of (x terms. The (r expansion of (x y)n has n + 1 b2 - 4ac is called the discriminant and determines the number and type of solutions. more➤ is the equation of a circle with radius r and center at 1h, k2. SOLVING NONLINEAR SYSTEMS A nonlinear system contains multivariable terms whose degrees are greater than one. A nonlinear system can be solved by the substitution method, the elimination method, or a combination of the two. 1)st term of the binomial y)n for r 0, 1, …, n is 4 5 Given an equation of a circle in general form, complete the squares on the x and y terms separately to put the equation into center-radius form. more➤ n! xn - ry r. r!1n - r2! 6 SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 2 Algebra Review Systems of Linear Equations (continued) Elimination Method i. Write the equations in standard form: Inequalities and Absolute Value: One Variable Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an inequality without changing the solution. ii. Multiplication by positive numbers: Each side of an inequality may be multiplied (or divided) by the same positive number without changing the solution. iii. Multiplication by negative numbers: If each side of an inequality is multiplied (or divided) by the same negative number, the direction of the inequality symbol is reversed. Inequalities and Absolute Value: One Variable (continued) Graphing a Linear Inequality i. If the inequality sign is replaced by an equals sign, the resulting line is the equation of the boundary. ii. Draw the graph of the boundary line, making the line solid if the inequality involves … or Ú or dashed if the inequality involves < or >. iii. Choose any point not on the line as a test point and substitute its coordinates into the inequality. iv. If the test point satisfies the inequality, shade the region that includes the test point; otherwise, shade the region that does not include the test point. Roots and Radicals n n Ax + By = C. ii. Multiply one or both equations by appropriate numbers so that the sum of the coefficient of one variable is 0. iii. Add the equations to eliminate one of the variables. iv. Solve the equation that results from step iii. v. Substitute the solution from step iv into either of the original equations to find the value of the remaining variable. Notes: If the result of step iii is a false statement, the graphs are parallel lines and there is no solution. If the result of step iii is a true statement, such as 0 = 0, the graphs are the same line, and the solution is every ordered pair on either line (of which there are infinitely many). 2an = a if n is odd. Rational Exponents n n a1>n: If 1 a is real, then a1>n = 1 a. 0 n n 2a is the principal or positive nth root of a. - 2 a is the negative nth root of a. 2a = |a| if n is even. n 2a = b means bn = a. Roots and Radicals The imaginary unit is i = 2 - 1, so i 2 = - 1. Radical Expressions and Graphs (continued) Quadratic Equations, Inequalities, and Functions (continued) Discriminant Number and Type of Solution Two real solutions One real solution Two complex solutions Inverse, Exponential, and Logarithmic Functions (continued) For a 7 0, a Z 1, f1x2 = ax defines the exponential function with base a. Properties of the graph of f1x2 = ax: i. Contains the point (0, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The x-axis is an asymptote. iv. Domain: ( - q , q ); Range: (0, q ) Conic Sections and Nonlinear Systems (continued) Sequences and Series A sequence is a list of terms t1, t2, t3, … (finite or infinite) whose general (nth) term is denoted tn. A series is the sum of the terms in a sequence. For b 7 0, 2 - b = i 2b. To multiply rad- COMPLEX NUMBERS Exponential Functions each factor to the form i 2b. A complex number has the form a + bi, where a and b are real numbers. ELLIPSE Equation of an Ellipse (Standard Position, Major Axis along x-axis) y x + 2 = 1, a 7 b 7 0 a2 b is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1a, 02 and 1 - a, 02 and y-intercepts are 10, b2 10, - b2. Foci are 1c, 02 and 1 - c, 02, 2 2 where c = 2a - b . 2 2 icals with negative radicands, first change b2 - 4ac 7 0 b2 - 4ac = 0 b2 - 4ac 6 0 ARITHMETIC SEQUENCES An arithmetic sequence is a sequence in which the difference between successive terms is a constant. Let a1 be the first term, an be the n th term, and d be the common difference. Common difference: d = an+1 – an nth term: an = a1 + 1n - 12d QUADRATIC FUNCTIONS Standard Form f1x2 = ax2 + bx + c, for a, b, c real, a Z 0. The graph is a parabola, opening up if a : If m and n are positive integers with m/n in lowest terms and a1>n is real, then If a1>n is not real, then am>n is not real. m>n OPERATIONS ON COMPLEX NUMBERS Adding and Subtracting Complex Numbers Add (or subtract) the real parts and add (or subtract) the imaginary parts. Solving Linear Inequalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. (Reverse the inequality symbol when multiplying or dividing by a negative number.) Functions Function Notation A function is a set of ordered pairs (x, y) such that for each first component x, there is one and only one second component y. The set of first components is called the domain, and the set of second components is called the range. y = f(x) defines y as a function of x. To write an equation that defines y as a function of x in function notation, solve the equation for y and replace y by f (x). To evaluate a function written in function notation for a given value of x, substitute the value wherever x appears. Product Rule: If 1a and 1b are real and n is a natural number, then am>n = 1a1>n2m. a 7 0, down if a 6 0. The vertex is b a -a , 4ac 42 a b2 Logarithmic Functions The logarithmic function is the inverse of the exponential function: y = loga x means x = ay. For a 7 0, a Z 1, g1x2 = loga x defines the logarithmic function with base a. Properties of the graph of g1x2 = loga x: i. Contains the points (1, 0) and (a, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The y-axis is an asymptote. iv. Domain: (0, q ); Range: ( - q , q ) SIMPLIFYING RADICAL EXPRESSIONS Quotient Rule: If 1a and 1b are real and n is a natural number, then n n n n n Solving Compound Inequalities i. Solve each inequality in the compound, inequality individually. ii. If the inequalities are joined with and, then the solution set is the intersection of the two individual solution sets. iii. If the inequalities are joined with or, then the solution set is the union of the two individual solution sets. 1a # 1b = 1ab. a 1a =n. Ab 1b n n n n b. -b 2a . Sum of the first n terms: Multiplying Complex Numbers Multiply using FOIL expansion and using i 2 = - 1 to reduce the result. The axis of symmetry is x = Equation of an Ellipse (Standard Position, Major Axis along y-axis) y2 a 2 Sn = Dividing Complex Numbers Multiply the numerator and the denominator by the conjugate of the denominator. THREE VARIABLES i. Use the elimination method to eliminate any variable from any two of the original equations. ii. Eliminate the same variable from any other two equations. iii. Steps i and ii produce a system of two equations in two variables. Use the elimination method for two-variable systems to solve for the two variables. iv. Substitute the values from step iii into any of the original equations to find the value of the remaining variable. Vertex Form f1x2 = a1x - h22 + k. The vertex is 1h, k2. The axis of symmetry is x = h. Horizontal Parabola The graph of x = ay2 + by + c, is a horizontal parabola, opening to the right if + x2 = 1, a 7 b 7 0 b2 n n 1a1 + an2 = 32a1 + 1n - 12d4 2 2 GEOMETRIC SEQUENCES A geometric sequence is a sequence in which the ratio of successive terms is a constant. Let t1 be the first term, t n be the nth term, and r be the common ratio. Common ratio: r = OPERATIONS ON RADICAL EXPRESSIONS Adding and Subtracting: Only radical expressions with the same index and the same radicand can be combined. Multiplying: Multiply binomial radical expressions by using FOIL expansion. Dividing: Rationalize the denominator by multiplying both the numerator and denominator by the same expression. If the denominator involves the sum of an integer and a square root, the expression used will be chosen to create a difference of squares. Quadratic Equations, Inequalities, and Functions If a is a complex number, then the solutions to x2 = a are x = 1a and x = - 1a. a 7 0, to the left if a 6 0. Note that this is not the graph of a function. is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1b, 02 and 1 - b, 02 and y-intercepts are 10, a2 10, - a2. Foci are 10, c2 and 10, - c2, where c = 2a2 - b2. tn + 1 tn SOLVING QUADRATIC EQUATIONS Square Root Property QUADRATIC INEQUALITIES Solving Quadratic (or HigherDegree Polynomial) Inequalities i. Replace the inequality sign by an equality sign and find the real-valued solutions to the equation. ii. Use the solutions from step i to divide the real number line into intervals. iii. Substitute a test number from each interval into the original inequality to determine the intervals that belong to the solution set. iv. Consider the endpoints separately. Solving Absolute Value Equations and Inequalities Suppose k is positive. To solve ƒ ax + b ƒ = k, solve the compound equation ax + b = k or ax + b = - k. To solve ƒ ax + b ƒ 7 k, solve the compound inequality ax + b 7 k or ax + b 6 - k. To solve ƒ ax + b ƒ 6 k, solve the compound inequality - k 6 ax + b 6 k. To solve an absolute value equation of the form ƒ ax + b ƒ = ƒ cx + d ƒ , solve the compound equation ax + b = cx + d or Variation If there exists some real number (constant) k such that: y = kx n, then y varies directly as xn. APPLICATIONS i. Assign variables to the unknown quantities in the problem. ii. Write a system of equations that relates the unknowns. iii. Solve the system. Solving Quadratic Equations by Completing the Square To solve ax2 + bx + c = 0, a Z 0: i. If a Z 1, divide each side by a. ii. Write the equation with the variable terms on one side of the equals sign and the constant on the other. iii. Take half the coefficient of x and square it. Add the square to each side of the equation. iv. Factor the perfect square trinomial and write it as the square of a binomial. Combine the constants on the other side. v. Use the square root property to determine the solutions. MATRIX ROW OPERATIONS i. Any two rows of the matrix may be interchanged. ii. All the elements in any row may be multiplied by any nonzero real number. iii. Any row may be modified by adding to the elements of the row the product of a real number and the elements of another row. A system of equations can be represented by a matrix and solved by matrix methods. Write an augmented matrix and use row operations to reduce the matrix to row echelon form. k , then y varies inversely as xn. xn y = kxz, then y varies jointly as x and z. Operations on Functions If f(x) and g(x) are functions, then the y= following functions are derived from f and g: Solving Equations Involving Radicals i. Isolate one radical on one side of the equation. ii. Raise both sides of the equation to a power that equals the index of the radical. iii. Solve the resulting equation; if it still contains a radical, repeat steps i and ii. iv. The resulting solutions are only candidates. Check which ones satisfy the original equation. Candidates that do not check are extraneous (not part of the solution set). more➤ 1f - g21x2 = f1x2 - g1x2 f 1x2 f a b (x) = , g1x2 Z 0 g g1x2 Composition of f and g: 1f + g21x2 = f1x2 + g1x2 1fg21x2 = f1x2 # g1x2 Inverse, Exponential, and Logarithmic Functions Inverse Functions If any horizontal line intersects the graph of a function in, at most, one point, then the function is one to one and has an inverse. If y = f (x) is one to one, then the equation that defines the inverse function f –1 is found by interchanging x and y, solving for y, and replacing y with f –1(x). The graph of f –1 is the mirror image of the graph of f with respect to the line y = x . more➤ Logarithm Rules Product rule: log a xy = log a x + log a y x Quotient rule: log a y = log a x - log a y r Power rule: log a x = r log a x Special properties: a log a x = x, log a ax = x Change-of-base rule: For a 7 0, a Z 1, logb x b 7 0, b Z 1, x 7 0, log a x = . logb a Exponential, Logarithmic Equations Suppose b 7 0, b Z 1. i. If bx = by, then x = y. ii. If x 7 0, y 7 0, then log b x = log b y is equivalent to x = y. iii. If log b x = y, then by = x. HYPERBOLA Equation of a Hyperbola (Standard Position, Opening Left and Right) y2 x2 - 2=1 2 a b 1a, 02 and 1 - a, 02. Foci are 1c, 02 and 1 - c, 02, where c = 2a2 + b2. b Asymptotes are y = ; a x. Equation of a Hyperbola (Standard Position, Opening Up and Down) y2 a 2 nth term: tn = t1r n - 1 Sum of the first n terms: Sn = t11r n - 12 r-1 ,r Z 1 is the equation of a hyperbola centered at Sum of the terms of an infinite geometric sequence with |r| < 1: S = the origin, whose x-intercepts (vertices) are t1 1-r The Binomial Theorem Factorials For any positive integer n, and - x2 b2 =1 n! = n1n - 121n - 22 Á 132122112 0! = 1. Binomial Coefficient Conic Sections and Nonlinear Systems CIRCLE Equation of a Circle: Center-Radius 1x - h22 + 1y - k22 = r2 Equation of a Circle: General x2 + y2 + ax + by + c = 0 is the equation of a hyperbola centered at the where c = 2a2 + b2. Asymptotes are a y = ; b x. origin, whose y-intercepts (vertices) are 10, a2 and 10, - a2. Foci are 10, c2 and 10, - c2, ax + b = - 1cx + d2. more➤ Quadratic Formula The solutions of ax2 + bx + c = 0, a Z 0 are given by x= -b 1f g21x2 = f 3g1x24 2b2 - 4ac . 2a For any nonnegative integers n and r, with n n! . r … n, a b = nCp = r r!1n - r2! The binomial expansion of (x terms. The (r expansion of (x y)n has n + 1 b2 - 4ac is called the discriminant and determines the number and type of solutions. more➤ is the equation of a circle with radius r and center at 1h, k2. SOLVING NONLINEAR SYSTEMS A nonlinear system contains multivariable terms whose degrees are greater than one. A nonlinear system can be solved by the substitution method, the elimination method, or a combination of the two. 1)st term of the binomial y)n for r 0, 1, …, n is 4 5 Given an equation of a circle in general form, complete the squares on the x and y terms separately to put the equation into center-radius form. more➤ n! xn - ry r. r!1n - r2! 6 SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 2 Algebra Review Systems of Linear Equations (continued) Elimination Method i. Write the equations in standard form: Inequalities and Absolute Value: One Variable Properties i. Addition: The same quantity may be added to (or subtracted from) each side of an inequality without changing the solution. ii. Multiplication by positive numbers: Each side of an inequality may be multiplied (or divided) by the same positive number without changing the solution. iii. Multiplication by negative numbers: If each side of an inequality is multiplied (or divided) by the same negative number, the direction of the inequality symbol is reversed. Inequalities and Absolute Value: One Variable (continued) Graphing a Linear Inequality i. If the inequality sign is replaced by an equals sign, the resulting line is the equation of the boundary. ii. Draw the graph of the boundary line, making the line solid if the inequality involves … or Ú or dashed if the inequality involves < or >. iii. Choose any point not on the line as a test point and substitute its coordinates into the inequality. iv. If the test point satisfies the inequality, shade the region that includes the test point; otherwise, shade the region that does not include the test point. Roots and Radicals n n Ax + By = C. ii. Multiply one or both equations by appropriate numbers so that the sum of the coefficient of one variable is 0. iii. Add the equations to eliminate one of the variables. iv. Solve the equation that results from step iii. v. Substitute the solution from step iv into either of the original equations to find the value of the remaining variable. Notes: If the result of step iii is a false statement, the graphs are parallel lines and there is no solution. If the result of step iii is a true statement, such as 0 = 0, the graphs are the same line, and the solution is every ordered pair on either line (of which there are infinitely many). 2an = a if n is odd. Rational Exponents n n a1>n: If 1 a is real, then a1>n = 1 a. 0 n n 2a is the principal or positive nth root of a. - 2 a is the negative nth root of a. 2a = |a| if n is even. n 2a = b means bn = a. Roots and Radicals The imaginary unit is i = 2 - 1, so i 2 = - 1. Radical Expressions and Graphs (continued) Quadratic Equations, Inequalities, and Functions (continued) Discriminant Number and Type of Solution Two real solutions One real solution Two complex solutions Inverse, Exponential, and Logarithmic Functions (continued) For a 7 0, a Z 1, f1x2 = ax defines the exponential function with base a. Properties of the graph of f1x2 = ax: i. Contains the point (0, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The x-axis is an asymptote. iv. Domain: ( - q , q ); Range: (0, q ) Conic Sections and Nonlinear Systems (continued) Sequences and Series A sequence is a list of terms t1, t2, t3, … (finite or infinite) whose general (nth) term is denoted tn. A series is the sum of the terms in a sequence. For b 7 0, 2 - b = i 2b. To multiply rad- COMPLEX NUMBERS Exponential Functions each factor to the form i 2b. A complex number has the form a + bi, where a and b are real numbers. ELLIPSE Equation of an Ellipse (Standard Position, Major Axis along x-axis) y x + 2 = 1, a 7 b 7 0 a2 b is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1a, 02 and 1 - a, 02 and y-intercepts are 10, b2 10, - b2. Foci are 1c, 02 and 1 - c, 02, 2 2 where c = 2a - b . 2 2 icals with negative radicands, first change b2 - 4ac 7 0 b2 - 4ac = 0 b2 - 4ac 6 0 ARITHMETIC SEQUENCES An arithmetic sequence is a sequence in which the difference between successive terms is a constant. Let a1 be the first term, an be the n th term, and d be the common difference. Common difference: d = an+1 – an nth term: an = a1 + 1n - 12d QUADRATIC FUNCTIONS Standard Form f1x2 = ax2 + bx + c, for a, b, c real, a Z 0. The graph is a parabola, opening up if a : If m and n are positive integers with m/n in lowest terms and a1>n is real, then If a1>n is not real, then am>n is not real. m>n OPERATIONS ON COMPLEX NUMBERS Adding and Subtracting Complex Numbers Add (or subtract) the real parts and add (or subtract) the imaginary parts. Solving Linear Inequalities i. Simplify each side separately. ii. Isolate the variable term on one side. iii. Isolate the variable. (Reverse the inequality symbol when multiplying or dividing by a negative number.) Functions Function Notation A function is a set of ordered pairs (x, y) such that for each first component x, there is one and only one second component y. The set of first components is called the domain, and the set of second components is called the range. y = f(x) defines y as a function of x. To write an equation that defines y as a function of x in function notation, solve the equation for y and replace y by f (x). To evaluate a function written in function notation for a given value of x, substitute the value wherever x appears. Product Rule: If 1a and 1b are real and n is a natural number, then am>n = 1a1>n2m. a 7 0, down if a 6 0. The vertex is b a -a , 4ac 42 a b2 Logarithmic Functions The logarithmic function is the inverse of the exponential function: y = loga x means x = ay. For a 7 0, a Z 1, g1x2 = loga x defines the logarithmic function with base a. Properties of the graph of g1x2 = loga x: i. Contains the points (1, 0) and (a, 1) ii. If a 7 1, the graph rises from left to right. If 0 6 a 6 1, the graph falls from left to right. iii. The y-axis is an asymptote. iv. Domain: (0, q ); Range: ( - q , q ) SIMPLIFYING RADICAL EXPRESSIONS Quotient Rule: If 1a and 1b are real and n is a natural number, then n n n n n Solving Compound Inequalities i. Solve each inequality in the compound, inequality individually. ii. If the inequalities are joined with and, then the solution set is the intersection of the two individual solution sets. iii. If the inequalities are joined with or, then the solution set is the union of the two individual solution sets. 1a # 1b = 1ab. a 1a =n. Ab 1b n n n n b. -b 2a . Sum of the first n terms: Multiplying Complex Numbers Multiply using FOIL expansion and using i 2 = - 1 to reduce the result. The axis of symmetry is x = Equation of an Ellipse (Standard Position, Major Axis along y-axis) y2 a 2 Sn = Dividing Complex Numbers Multiply the numerator and the denominator by the conjugate of the denominator. THREE VARIABLES i. Use the elimination method to eliminate any variable from any two of the original equations. ii. Eliminate the same variable from any other two equations. iii. Steps i and ii produce a system of two equations in two variables. Use the elimination method for two-variable systems to solve for the two variables. iv. Substitute the values from step iii into any of the original equations to find the value of the remaining variable. Vertex Form f1x2 = a1x - h22 + k. The vertex is 1h, k2. The axis of symmetry is x = h. Horizontal Parabola The graph of x = ay2 + by + c, is a horizontal parabola, opening to the right if + x2 = 1, a 7 b 7 0 b2 n n 1a1 + an2 = 32a1 + 1n - 12d4 2 2 GEOMETRIC SEQUENCES A geometric sequence is a sequence in which the ratio of successive terms is a constant. Let t1 be the first term, t n be the nth term, and r be the common ratio. Common ratio: r = OPERATIONS ON RADICAL EXPRESSIONS Adding and Subtracting: Only radical expressions with the same index and the same radicand can be combined. Multiplying: Multiply binomial radical expressions by using FOIL expansion. Dividing: Rationalize the denominator by multiplying both the numerator and denominator by the same expression. If the denominator involves the sum of an integer and a square root, the expression used will be chosen to create a difference of squares. Quadratic Equations, Inequalities, and Functions If a is a complex number, then the solutions to x2 = a are x = 1a and x = - 1a. a 7 0, to the left if a 6 0. Note that this is not the graph of a function. is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1b, 02 and 1 - b, 02 and y-intercepts are 10, a2 10, - a2. Foci are 10, c2 and 10, - c2, where c = 2a2 - b2. tn + 1 tn SOLVING QUADRATIC EQUATIONS Square Root Property QUADRATIC INEQUALITIES Solving Quadratic (or HigherDegree Polynomial) Inequalities i. Replace the inequality sign by an equality sign and find the real-valued solutions to the equation. ii. Use the solutions from step i to divide the real number line into intervals. iii. Substitute a test number from each interval into the original inequality to determine the intervals that belong to the solution set. iv. Consider the endpoints separately. Solving Absolute Value Equations and Inequalities Suppose k is positive. To solve ƒ ax + b ƒ = k, solve the compound equation ax + b = k or ax + b = - k. To solve ƒ ax + b ƒ 7 k, solve the compound inequality ax + b 7 k or ax + b 6 - k. To solve ƒ ax + b ƒ 6 k, solve the compound inequality - k 6 ax + b 6 k. To solve an absolute value equation of the form ƒ ax + b ƒ = ƒ cx + d ƒ , solve the compound equation ax + b = cx + d or Variation If there exists some real number (constant) k such that: y = kx n, then y varies directly as xn. APPLICATIONS i. Assign variables to the unknown quantities in the problem. ii. Write a system of equations that relates the unknowns. iii. Solve the system. Solving Quadratic Equations by Completing the Square To solve ax2 + bx + c = 0, a Z 0: i. If a Z 1, divide each side by a. ii. Write the equation with the variable terms on one side of the equals sign and the constant on the other. iii. Take half the coefficient of x and square it. Add the square to each side of the equation. iv. Factor the perfect square trinomial and write it as the square of a binomial. Combine the constants on the other side. v. Use the square root property to determine the solutions. MATRIX ROW OPERATIONS i. Any two rows of the matrix may be interchanged. ii. All the elements in any row may be multiplied by any nonzero real number. iii. Any row may be modified by adding to the elements of the row the product of a real number and the elements of another row. A system of equations can be represented by a matrix and solved by matrix methods. Write an augmented matrix and use row operations to reduce the matrix to row echelon form. k , then y varies inversely as xn. xn y = kxz, then y varies jointly as x and z. Operations on Functions If f(x) and g(x) are functions, then the y= following functions are derived from f and g: Solving Equations Involving Radicals i. Isolate one radical on one side of the equation. ii. Raise both sides of the equation to a power that equals the index of the radical. iii. Solve the resulting equation; if it still contains a radical, repeat steps i and ii. iv. The resulting solutions are only candidates. Check which ones satisfy the original equation. Candidates that do not check are extraneous (not part of the solution set). more➤ 1f - g21x2 = f1x2 - g1x2 f 1x2 f a b (x) = , g1x2 Z 0 g g1x2 Composition of f and g: 1f + g21x2 = f1x2 + g1x2 1fg21x2 = f1x2 # g1x2 Inverse, Exponential, and Logarithmic Functions Inverse Functions If any horizontal line intersects the graph of a function in, at most, one point, then the function is one to one and has an inverse. If y = f (x) is one to one, then the equation that defines the inverse function f –1 is found by interchanging x and y, solving for y, and replacing y with f –1(x). The graph of f –1 is the mirror image of the graph of f with respect to the line y = x . more➤ Logarithm Rules Product rule: log a xy = log a x + log a y x Quotient rule: log a y = log a x - log a y r Power rule: log a x = r log a x Special properties: a log a x = x, log a ax = x Change-of-base rule: For a 7 0, a Z 1, logb x b 7 0, b Z 1, x 7 0, log a x = . logb a Exponential, Logarithmic Equations Suppose b 7 0, b Z 1. i. If bx = by, then x = y. ii. If x 7 0, y 7 0, then log b x = log b y is equivalent to x = y. iii. If log b x = y, then by = x. HYPERBOLA Equation of a Hyperbola (Standard Position, Opening Left and Right) y2 x2 - 2=1 2 a b 1a, 02 and 1 - a, 02. Foci are 1c, 02 and 1 - c, 02, where c = 2a2 + b2. b Asymptotes are y = ; a x. Equation of a Hyperbola (Standard Position, Opening Up and Down) y2 a 2 nth term: tn = t1r n - 1 Sum of the first n terms: Sn = t11r n - 12 r-1 ,r Z 1 is the equation of a hyperbola centered at Sum of the terms of an infinite geometric sequence with |r| < 1: S = the origin, whose x-intercepts (vertices) are t1 1-r The Binomial Theorem Factorials For any positive integer n, and - x2 b2 =1 n! = n1n - 121n - 22 Á 132122112 0! = 1. Binomial Coefficient Conic Sections and Nonlinear Systems CIRCLE Equation of a Circle: Center-Radius 1x - h22 + 1y - k22 = r2 Equation of a Circle: General x2 + y2 + ax + by + c = 0 is the equation of a hyperbola centered at the where c = 2a2 + b2. Asymptotes are a y = ; b x. origin, whose y-intercepts (vertices) are 10, a2 and 10, - a2. Foci are 10, c2 and 10, - c2, ax + b = - 1cx + d2. more➤ Quadratic Formula The solutions of ax2 + bx + c = 0, a Z 0 are given by x= -b 1f g21x2 = f 3g1x24 2b2 - 4ac . 2a For any nonnegative integers n and r, with n n! . r … n, a b = nCp = r r!1n - r2! The binomial expansion of (x terms. The (r expansion of (x y)n has n + 1 b2 - 4ac is called the discriminant and determines the number and type of solutions. more➤ is the equation of a circle with radius r and center at 1h, k2. SOLVING NONLINEAR SYSTEMS A nonlinear system contains multivariable terms whose degrees are greater than one. A nonlinear system can be solved by the substitution method, the elimination method, or a combination of the two. 1)st term of the binomial y)n for r 0, 1, …, n is 4 5 Given an equation of a circle in general form, complete the squares on the x and y terms separately to put the equation into center-radius form. more➤ n! xn - ry r. r!1n - r2! 6 ...
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