Chapter 6 Factoring and Quadratic Equations Section 1

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Chapter 6 Factoring and Quadratic Equations Section 1 An Introduction to Factoring; the Greatest

Chapter 6 Factoring and Quadratic Equations Section 1 An Introduction to Factoring; the Greatest Common Factor; Factoring by Grouping Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 1

Study Strategy Study Groups üUnderstanding Copyright © 2016, 2012, and 2009 Pearson Education, Inc.

Study Strategy Study Groups üUnderstanding Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 2

Concept Greatest Common Factor (GCF) The greatest common factor of two or more integers

Concept Greatest Common Factor (GCF) The greatest common factor of two or more integers is the largest whole number that is a factor of each integer. For a variable factor to be included in the GCF, it must be a factor of each term. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 3

Example Find the GCF of 48 and 54. 1 • The prime factorization of

Example Find the GCF of 48 and 54. 1 • The prime factorization of 48 is 24 · 3 and the prime factorization of 54 is 33 · 2. The prime factors that they share 2 and 3, which is the smallest power of each. The GCF is 2 · 3 = 6. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 4

Example Find the GCF of 384, 420 and 504. 2 The prime factorizations: 384

Example Find the GCF of 384, 420 and 504. 2 The prime factorizations: 384 = 27 · 3 420 = 22 · 3 · 5 · 7 504 = 23 · 32 · 7 The only primes that are factors of all three numbers are 2 and 3. The smallest power of 2 is 22, and the smallest power of 3 is 31. The GCF is 22 · 3 = 12. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 5

Example Greatest Common Factor (GCF) 3 Find the GCF: GCF of 35, 14, &

Example Greatest Common Factor (GCF) 3 Find the GCF: GCF of 35, 14, & 49: 7 GCF: Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 6

Concept Factoring Out the Greatest Common Factor The first step for factoring any polynomial

Concept Factoring Out the Greatest Common Factor The first step for factoring any polynomial is to factor out the GCF of all of the terms. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 7

Example 4 Factoring Out the Greatest Common Factor out the GCF: Copyright © 2016,

Example 4 Factoring Out the Greatest Common Factor out the GCF: Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 8

Example 5 Factoring Out the Greatest Common Factor out the GCF: Copyright © 2016,

Example 5 Factoring Out the Greatest Common Factor out the GCF: Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 9

Example Factor GCF. by factoring out the 6 The binomial b + 5 is

Example Factor GCF. by factoring out the 6 The binomial b + 5 is a common factor for these two terms. Begin by factoring out this common factor. The only factors that remain are a and 8. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 10

Concept Factoring by Grouping To factor a polynomial with four terms by grouping, we

Concept Factoring by Grouping To factor a polynomial with four terms by grouping, we split the polynomial into two groups. We factor a common factor out of the first two terms and another common factor out of the last two terms. If the two “groups” share a common binomial factor, then this binomial can then be factored out to complete the factoring of the polynomial. Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 11

Example 7 Factoring by Grouping Factor by grouping: Copyright © 2016, 2012, and 2009

Example 7 Factoring by Grouping Factor by grouping: Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 12

Example 8 Factoring by Grouping Factor by grouping: Copyright © 2016, 2012, and 2009

Example 8 Factoring by Grouping Factor by grouping: Copyright © 2016, 2012, and 2009 Pearson Education, Inc. 13