8.4
Multiplying and Dividing Radicals
Product Rule for Radicals
If
n
a and n b are real numbers,
n
a n b = n ab
Martin-Gay, Beginning Algebra, 5ed
2
Multiplying Radical Expressions
Example: Simplify the following radical expressions.
3 y 5
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
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
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
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
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
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
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
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
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
6.4
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
6.3
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
6.2
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
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
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
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
5.4
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
5.3
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
3.3
Intercepts
Intercepts
Intercepts of axes (where graph crosses the axes) Since all points on the x-axis have a y-coordinate of 0, to find x-intercept, let y = 0 and solve for x Since all points on the y-axis have an x-coordinate of 0, to find y
3.2
Graphing Linear Equations
Linear Equations
Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form Ax + By = C where A, B, and C are real numbers and A and B not both 0. The graph of a
Chapter 3
Graphing
3.1
Reading Graphs and the Rectangular Coordinate System
Bar Graphs
A bar graph consists of a series of bars arranged vertically or horizontally. The following bar graph shows the attendance at each Super Bowl game from 2000 t
2.9
Solving Linear Inequalities
Linear Inequalities in One Variable
A linear inequality in one variable is an inequality that can be written in the form ax + b < c where a, b, and c are real numbers and a is not 0.
This definition and all other d
2.8
Further 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 possible Pr
2.7
Percent and Mixture 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
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
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
2.4
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
2.3
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