CHAPTER

6Polynomials: Factoring

(continued)

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6.1 Multiplying and Simplifying Rational Expressions

6.2 Division and Reciprocals

6.3 Least Common Multiples and Denominators

6.4 Adding Rational Expressions

6.5 Subtracting Rational Expressions

6.6 Solving Rational Equations

6.7 Applications Using Rational Equations and Proportions

CHAPTER

6Polynomials: Factoring

Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

6.8 Complex Rational Expressions

6.9 Direct Variation and Inverse Variation

OBJECTIVES

6.3 Least Common Multiples and Denominators

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a Find the LCM of several numbers by factoring.b Add fractions, first finding the LCD.c Find the LCM of algebraic expressions by factoring

Least Common Multiples To add when denominators are different, we first find a common denominator.

To find the LCM of 12 and 30, we factor:12 = 2 · 2 · 330 = 2 · 3 · 5

The LCM is the number that has 2 as a factor twice, 3 as a factor once, and 5 as a factor once:

LCM = 2 · 2 · 3 · 5 = 60

6.3 Least Common Multiples and Denominators

a Find the LCM of several numbers by factoring.

Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

To find the LCM, use each factor the greatest number of times that is appears in any one factorization.

6.3 Least Common Multiples and Denominators

Finding LCMs

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EXAMPLESolution

48 = 2 · 2 · 2 · 2 · 3

54 = 2 · 3 · 3 · 3

LCM = 2 2 2 2 3 3 3 or 432

6.3 Least Common Multiples and Denominators

a Find the LCM of several numbers by factoring.

A Find the LCM of 48 and 54.

Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE7 15

.18 24

18 32 3

2 24 322 2 2 2 3 3LCD or 72,

7 15

18 24

7 15

2 3 3 2 2

4

4 32

3

3

28 45

2 2 2 3 3

73

72

Solution

6.3 Least Common Multiples and Denominators

b Add fractions, first finding the LCD.

B Add:

Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution

6x2 = 2 3 x x4x3 = 2 2 x x x

LCM = 2 2 3 x x x

The LCM = 22 3 x3, or 12x3.

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

C Find the LCM of 6x2 and 4x3

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EXAMPLE

a) 16a and 24b b) 24x4y4 and 6x6y2 c) x2 4 and x2 2x 8

Solutiona) 16a = 2 2 2 2 a 24b = 2 2 2 3 b

The LCM = 2 2 2 2 a 3 b

The LCM is 24 3 a b, or 48ab

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

D For each pair of polynomials, find the least common multiple.

(continued)

Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

b) 24x4y4 = 2 · 2 · 2 · 3 · x · x · x · x · y · y · y · y 6x6y2 = 2 · 3 · x · x · x · x · x · x · y · y

LCM = 2 · 2 · 2 · 3 · x · x · x · x · y · y · y · y · x · x

Note that we used the highest power of each factor. The LCM is 24x6y4

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

D Finding the LCM by Factoring

(continued)

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EXAMPLEc) x2 – 4 = (x – 2)(x + 2) x2 – 2x – 8 = (x + 2)(x – 4)

LCM = (x – 2)(x + 2)(x – 4)

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

D Finding the LCM by Factoring

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EXAMPLE

a) 15x, 30y, 25xyz b) x2 + 3, x + 2, 7Solutiona) 15x = 3 5 x 30y = 2 3 5 y 25xyz = 5 5 x y z

LCM = 2 3 5 5 x y zThe LCM is 2 3 52 x y z or 150xyz

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

E For each group of polynomials, find the least common multiple.

(continued)

Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

a) 15x, 30y, 25xyz b) x2 + 3, x + 2, 7Solution b) x2 + 3 = x + 2 = 7 =b) Since x2 + 3, x + 2, and 7 are not factorable, the LCM is their product: 7(x2 + 3)(x + 2).

6.3 Least Common Multiples and Denominators

c Find the LCM of algebraic expressions by factoring

E For each group of polynomials, find the least common multiple.

Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.