1.4
Introduction to Variable Expressions and Equations
Exponents
Exponential notation is used to write repeated multiplication in a more compact form.
3 3 3 3 3
3
4
4
exponent base
The expression 34 is called an exponential expression. Exam
5.4
Special Products
The FOIL Method
When multiplying two binomials, the distributive property can be easily remembered as the FOIL method. F product of First terms O product of Outside terms I product of Inside terms L product of Last terms
5.5
Negative Exponents and Scientific Notation
Negative Exponents
Negative Exponents If a is a real number other than 0 and n is an integer, then 1 n a n a Example:
Simplify by writing each expression with positive exponents.
1 1 3 = 2 = 3 9 2 R
5.6
Dividing Polynomials
Dividing Polynomials
Dividing a Polynomial by a Monomial
Divide each term of the polynomial by the monomial.
ab a b , c0 c c c
Example:
12a + 36a 15 12a 3 36a 15 = + 3a 3a 3a 3a 5 2 = 4a + 12 a
3
Martin-Gay, Beginn
Chapter 6
Factoring Polynomials
6.1
The Greatest Common Factor and Factor by Grouping
Factors
Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the ori
6.2
Factoring Trinomials of 2 the Form x + bx + c
Factoring Trinomials
Recall by using the FOIL method that
F O I L
(x + 2)(x + 4) = x2 + 4x + 2x + 8 = x2 + 6x + 8 To factor x2 + bx + c into (x + one #)(x + another #), note that b is the sum of t
6.3
Factoring Trinomials of the 2 Form ax + bx + c and Perfect Square Trinomials
Factoring Trinomials
Returning to the FOIL method, F O I L (3x + 2)(x + 4) = 3x2 + 12x + 2x + 8 = 3x2 + 14x + 8 To factor ax2 + bx + c into (#1x + #2)(#3x + #4), note
6.4
Factoring Trinomials of 2 the Form x + bx + c by Grouping
Factoring by Grouping
To Factor Trinomials by Grouping
Factor out a greatest common factor, if there is one other than 1. For the resulting trinomial ax2 + bx + c, find two numbers wh
6.5
Factoring Binomials
Difference of Two Squares
Another shortcut for factoring a trinomial is when we want to factor the difference of two squares. a2 b2 = (a + b)(a b) A binomial is the difference of two square if 1.both terms are squares and
6.6
Solving Quadratic Equations by Factoring
Zero Factor Theorem
Quadratic Equations
Can be written in the form ax2 + bx + c = 0. a, b and c are real numbers and a 0. This is referred to as standard form.
Zero Factor Theorem
If a and b are re
Chapter 7
Rational Expressions
7.1
Simplifying Rational Expressions
Rational Expressions
P Rational expressions can be written in the form Q
where P and Q are both polynomials and Q
0. Examples of Rational Expressions
3x 2 + 2 x 4 4x 5
4x
Chapter 7
Rational Expressions
Martin-Gay, Beginning Algebra, 5ed
7.1
Simplifying Rational Expressions
Martin-Gay, Beginning Algebra, 5ed
Rational Expressions
Rational expressionscan be written in the form where P and Q are both polynomials and
7.2
Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions
Multiplying rational expressions when P, Q, R, and S are polynomials with Q 0 and S 0.
P R PR = Q S QS
Martin-Gay, Beginning Algebra, 5ed
2
Multiplying Ratio
7.6
Proportions and Problem Solving with Rational Equations
Ratios and Rates
Ratio is the quotient of two numbers or two quantities. The ratio of the numbers a and b can also be a written as a:b, or . The units associated with the ratio are import
Chapter 8
Roots and Radicals
8.1
Introduction to Radicals
Square Roots
Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # th
8.2
Simplifying Radicals
Product Rule for Radicals
If
a and b are real numbers,
ab = a b
a a = if b b b 0
Martin-Gay, Beginning Algebra, 5ed
2
Simplifying Radicals
Example: Simplify the following radical expressions.
40 = 5 = 16 15 4 10 = 2
8.3
Adding and Subtracting Radicals
Sums and Differences
Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences.
a+b a + b a b a b
Martin-Gay, Begi
5.3
Multiplying Polynomials
Multiplying Polynomials
Multiplying polynomials
If all of the polynomials are monomials, use the associative and commutative properties. If any of the polynomials are not monomials, use the distributive property bef
5.2
Adding and Subtracting Polynomials
Polynomial Vocabulary
Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a number Polynomial a sum of terms involving
3.4
Slope and Rate of Change
Slope of a Line
Slope of a Line
rise y change y2 y1 m= = = run x change x2 x1 x2 x1
Martin-Gay, Beginning Algebra, 5ed
2
Slope of a Line
Slope of a Line
Positive Slope: m>0 y Lines with positive slopes go upward
1.5
Adding Real Numbers
Adding Real Numbers
Adding two numbers with the same sign
Add their absolute values. Use their common sign as the sign of the sum.
Add the following numbers.
3 + (5) = 8
Martin-Gay, Beginning Algebra, 5ed
2
Adding R
1.6
Subtracting Real Numbers
Subtracting Real Numbers
Subtracting real numbers
If a and b are real numbers, then a b = a + ( b).
Subtract the following numbers. ( 5) 6 ( 3) = ( 5) + ( 6) + 3 = 8
Martin-Gay, Beginning Algebra, 5ed
2
Complem
1.7
Multiplying and Dividing Real Numbers
Multiplying Real Numbers
The product of two numbers with the same sign is a positive number. Example: Multiply. 75 ( 3) 75 ( 3) = 225 Example: 5 2 Multiply. 12 3
5 12
The result will alway
1.8
Properties of Real Numbers
Commutative and Associative Property
Commutative property
Addition: a + b = b + a Multiplication: a b = b a
Associative property
Addition: (a + b) + c = a + (b + c)
Multiplication: (a b) c = a (b c)
Marti
Chapter 2
Equations, Inequalities and Problem Solving
2.1
Simplifying Algebraic Expressions
Terms
A term is a number, or the product of a number and variables raised to powers (the number is called a coefficient).
Examples of Terms
7 5x3 4xy2
2.2
The Addition Property of Equality
Linear Equations
Linear equations in one variable can be written in the form ax + b = c, where a, b and c are real numbers, and a 0. Equivalent equations are equations that have the same solution.
Martin-Gay
2.3
The Multiplication Property of Equality
Multiplication Property of Equality
Multiplication Property of Equality
If a, b, and c are real numbers, then a = b and ac = bc are equivalent equations Example: y=8 ( 1)( y) = 8( 1) y=8 Multiply both si
2.4
Solving Linear Equations
Solving Linear Equations
Solving Linear Equations in One Variable
1)
2)
3)
4)
5)
6)
Multiply both sides by the LCD to clear the equation of fractions if they occur. Use the distributive property to remove parenth
2.5
An Introduction to Problem Solving
Strategy for Problem Solving
General Strategy for Problem Solving
1)
2) 3) 4)
UNDERSTAND the problem Read and reread the problem Choose a variable to represent the unknown Construct a drawing, whenever p
2.6
Formulas and Problem Solving
Formulas
A formula is an equation that states a known relationship among multiple quantities (has more than one variable in it)
A = lw I = PRT P=a+b+c d = rt V = lwh (Area of a rectangle = length width) (Simple In