8-2
DESCRIPTION
Multiplying and Dividing Rational Expressions. 8-2. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 2. Holt Algebra 2. 1. y 2. x 6. y 5. y 3. x 2. Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2. x 7. 2. y 3 y 3. y 6. 3. - PowerPoint PPT PresentationTRANSCRIPT
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions8-2 Multiplying and Dividing Rational Expressions
Holt Algebra 2
Warm UpLesson PresentationLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Warm UpSimplify each expression. Assume all variables are nonzero.1. x5 x2
3. x7
x6
x2
Factor each expression.5. x2 – 2x – 8 6. x2 – 5x
(x – 4)(x + 2)
x3(x – 3)(x + 3)
2. y3 y3 y6
x4 y2
y54. 1
y3
7. x5 – 9x3
x(x – 5)
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Simplify rational expressions.Multiply and divide rational expressions.
Objectives
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
rational expressionVocabulary
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
In Lesson 8-1, you worked with inverse variation functions such as y = . The expression on the right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following:
5x
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.
Caution!
Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Simplify. Identify any x-values for which the expression is undefined.
Example 1A: Simplifying Rational Expressions
Quotient of Powers Property
10x8
6x4
510x8 – 4
3653
x4=
The expression is undefined at x = 0 because this value of x makes 6x4 equal 0.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Example 1B: Simplifying Rational Expressions
Simplify. Identify any x-values for which the expression is undefined.
x2 + x – 2 x2 + 2x – 3
(x + 2)(x – 1) (x – 1)(x + 3)
Factor; then divide out common factors.
= (x + 2)(x + 3)
The expression is undefined at x = 1 and x = –3 because these values of x make the factors (x – 1) and (x + 3) equal 0.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check Substitute x = 1 and x = –3 into the original expression.
(1)2 + (1) – 2 (1)2 + 2(1) – 3
00= (–3)2 + (–3) – 2
(–3)2 + 2(–3) – 340=
Both values of x result in division by 0, which is undefined.
Example 1B Continued
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 1a
Simplify. Identify any x-values for which the expression is undefined.
Quotient of Powers Property
16x11
8x2
28x11 – 2
18 2x9=
The expression is undefined at x = 0 because this value of x makes 8x2 equal 0.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 1b
Simplify. Identify any x-values for which the expression is undefined.
3x + 4 3x2 + x – 4
(3x + 4) (3x + 4)(x – 1)
Factor; then divide out common factors.
= 1(x – 1)
The expression is undefined at x = 1 and x = –because these values of x make the factors (x – 1) and (3x + 4) equal 0.
43
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
3(1) + 43(1)2 + (1) – 4
70=
Both values of x result in division by 0, which is undefined.
Check It Out! Example 1b Continued
Check Substitute x = 1 and x = – into the original expression.
43
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 1c
Simplify. Identify any x-values for which the expression is undefined.
6x2 + 7x + 2 6x2 – 5x – 5
(2x + 1)(3x + 2) (3x + 2)(2x – 3)
Factor; then divide out common factors.
= (2x + 1)(2x – 3)
The expression is undefined at x =– and x = because these values of x make the factors (3x + 2) and (2x – 3) equal 0.
32
23
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Both values of x result in division by 0, which is undefined.
Check Substitute x = and x = – into the original expression.
32
23
Check It Out! Example 1c Continued
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Simplify . Identify any x values for which the expression is undefined.
Example 2: Simplifying by Factoring by –1
Factor out –1 in the numerator so that x2 is positive, and reorder the terms.
Factor the numerator and denominator. Divide out common factors.
The expression is undefined at x = –2 and x = 4.
4x – x2 x2 – 2x – 8
–1(x2 – 4x)x2 – 2x – 8–1(x)(x – 4)
(x – 4)(x + 2)
–x(x + 2 ) Simplify.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check The calculator screens suggest that = except when x = – 2 or x = 4.
Example 2 Continued
4x – x2 x2 – 2x – 8
–x(x + 2)
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 2a
Simplify . Identify any x values for which the expression is undefined.
Factor out –1 in the numerator so that x is positive, and reorder the terms.
Factor the numerator and denominator. Divide out common factors.
The expression is undefined at x = 5.
10 – 2x x – 5
–1(2x – 10)x – 5
–2 1 Simplify.
–1(2)(x – 5)(x – 5)
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 2a Continued
Check The calculator screens suggest that = –2 except when x = 5.10 – 2x
x – 5
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 2b
Simplify . Identify any x values for which the expression is undefined.
Factor out –1 in the numerator so that x is positive, and reorder the terms.
Factor the numerator and denominator. Divide out common factors.
–x2 + 3x 2x2 – 7x + 3
–1(x2 – 3x)2x2 – 7x + 3
–x 2x – 1 Simplify.
–1(x)(x – 3)(x – 3)(2x – 1)
The expression is undefined at x = 3 and x = . 1 2
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 2b Continued
–x2 + 3x 2x2 – 7x + 3
Check The calculator screens suggest that = except when x = and x = 3.
–x 2x – 1
12
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
You can multiply rational expressions the same way that you multiply fractions.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Multiply. Assume that all expressions are defined.
Example 3: Multiplying Rational Expressions
A. 3x5y3 2x3y7 10x3y4
9x2y5
3x5y3 2x3y7 10x3y4
9x2y553
3
5x3 3y5
B. x – 3 4x + 20 x + 5
x2 – 9
x – 3 4(x + 5) x + 5
(x – 3)(x + 3)
1 4(x + 3)
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 3 Multiply. Assume that all expressions are defined.
A. x15 20
x4 2x x7
x15 20
x4 2x x7
3
22
2x3 3
B. 10x – 40 x2 – 6x + 8 x + 3
5x + 15
10(x – 4) (x – 4)(x – 2) x + 3
5(x + 3)
2 (x – 2)
2
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal.
1 2
3 4÷ = 1
2 4 3
2 2 3=
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Divide. Assume that all expressions are defined.
Example 4A: Dividing Rational Expressions
Rewrite as multiplication by the reciprocal.
5x4
8x2y2÷8y515
5x4
8x2y2 158y5
5x4
8x2y2 158y5
3
2 3
x2y3 3
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational ExpressionsExample 4B: Dividing Rational Expressions
x4 – 9x2 x2 – 4x + 3 ÷ x4 + 2x3 – 8x2
x2 – 16
Divide. Assume that all expressions are defined.
x4 – 9x2 x2 – 4x + 3 x2 – 16
x4 + 2x3 – 8x2Rewrite as multiplication by the reciprocal.
x2 (x2 – 9)x2 – 4x + 3 x2 – 16
x2(x2 + 2x – 8)x2(x – 3)(x + 3)(x – 3)(x – 1) (x + 4)(x – 4)
x2(x – 2)(x + 4)(x + 3)(x – 4) (x – 1)(x – 2)
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
x4 y
Check It Out! Example 4a
Rewrite as multiplication by the reciprocal.
x2
4 ÷ 12y2x4y
x2
4 12y2
2
3y x2
Divide. Assume that all expressions are defined.
x2
4 x4y12y2
3 1
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 4b
2x2 – 7x – 4 x2 – 9 ÷ 4x2– 1
8x2 – 28x +12
Divide. Assume that all expressions are defined.
(2x + 1)(x – 4)(x + 3)(x – 3) 4(2x2 – 7x + 3)
(2x + 1)(2x – 1)(2x + 1)(x – 4)(x + 3)(x – 3) 4(2x – 1)(x – 3)
(2x + 1)(2x – 1)4(x – 4) (x +3)
2x2 – 7x – 4 x2 – 9 8x2 – 28x +12
4x2– 1
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Example 5A: Solving Simple Rational Equations
Solve. Check your solution.
Note that x ≠ 5.
x2 – 25x – 5 = 14
(x + 5)(x – 5)(x – 5) = 14
x + 5 = 14x = 9
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
x2 – 25x – 5 = 14Check
(9)2 – 259 – 5 14 56
4 14
14 14
Example 5A Continued
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Example 5B: Solving Simple Rational Equations
Solve. Check your solution.
Note that x ≠ 2.
x2 – 3x – 10 x – 2 = 7
(x + 5)(x – 2)(x – 2) = 7
x + 5 = 7x = 2
Because the left side of the original equation is undefined when x = 2, there is no solution.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check A graphing calculator shows that 2 is not a solution.
Example 5B Continued
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 5a
Solve. Check your solution.
Note that x ≠ –4.
x2 + x – 12 x + 4 = –7
(x – 3)(x + 4)(x + 4) = –7
x – 3 = –7x = –4
Because the left side of the original equation is undefined when x = –4, there is no solution.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Check It Out! Example 5a Continued
Check A graphing calculator shows that –4 is not a solution.
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Solve. Check your solution.
4x2 – 92x + 3 = 5
(2x + 3)(2x – 3)(2x + 3) = 5
2x – 3 = 5x = 4
Check It Out! Example 5b
Note that x ≠ – .32
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
4x2 – 92x + 3 = 5Check
4(4)2 – 92(4) + 3 5
55 11 5
5 5
Check It Out! Example 5b Continued
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Lesson Quiz: Part I
1.
2.
Simplify. Identify any x-values for which the expression is undefined.
x2 – 6x + 5 x2 – 3x – 10
6x – x2
x2 – 7x + 6
x – 1 x + 2
x ≠ –2, 5
–x x – 1
x ≠ 1, 6
Holt McDougal Algebra 2
8-2 Multiplying and Dividing Rational Expressions
Lesson Quiz: Part II
3.
Multiply or divide. Assume that all expressions are defined.
x2 + 4x + 3x2 – 4 ÷ x2 + 2x – 3
x2 – 6x + 8 4.
x + 1 3x + 6 6x + 12
x2 – 1
(x + 1)(x – 4) (x + 2)(x – 1)
2 x – 1
5. x = 4
Solve. Check your solution.
4x2 – 12x – 1 = 9