1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 7.1 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 7.1 - 2 Factoring Chapter 7

3. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 7.1 - 3 7.1 Greatest Common Factors; Factoring by Grouping

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 4 7.1 Greatest Common Factors; Factoring by Grouping Objectives 1. Factor out the greatest common factor. 2. Factor by grouping.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 5 7.1 Greatest Common Factors; Factoring by Grouping Factor Out the Greatest Common Factor Just as we can multiply 2 and 4 to get 8, we can unmultiply 8 to obtain 2 and 4. We call such unmultiplication factoring. Similarly, we can sometimes factor a polynomial by writing it as the product of two or more simpler polynomials. Both processes use the distributive property. Multiplying “undoes” factoring, and factoring “undoes” multiplying.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 6 7.1 Greatest Common Factors; Factoring by Grouping Factoring Out the Greatest Common Factor The first step in factoring is to find the greatest common factor (GCF) – the largest term that divides each term of the polynomial. The GCD is a factor of all the terms of the polynomial.

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 7 7.1 Greatest Common Factors; Factoring by Grouping Factoring Out the Greatest Common Factor

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 8 7.1 Greatest Common Factors; Factoring by Grouping Factoring Out a Binomial Factor The greatest common factor need not be a monomial. Think of this as two terms with a common factor of (a + b). Think of this as two terms with a common factor of a(b – c) 2.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 9 7.1 Greatest Common Factors; Factoring by Grouping Factoring Out a Negative Common Factor When the coefficient of the term of greatest degree is negative, it is sometimes preferable to factor out the –1 that is understood along with the GCF. Factor only the 3 out. Or factor the – 3 out. Either is correct.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 10 7.1 Greatest Common Factors; Factoring by Grouping Factoring by Grouping Many polynomials have no greatest common factor other than the number 1. Some of these can be factored using the distributive property if those terms with a common factor are grouped together. Consider the polynomial: 1.The first two terms have a 5 in common, whereas, 2.The last two terms have an x in common. Applying the distributive property, we have

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 11 7.1 Greatest Common Factors; Factoring by Grouping Factoring by Grouping This last expression can be thought of as having two terms, Applying the distributive property again to factor (x + y) from each term: and.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 12 7.1 Greatest Common Factors; Factoring by Grouping Factoring by Grouping.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 13 7.1 Greatest Common Factors; Factoring by Grouping Factoring by Grouping Often there is more than one way to group. It does not matter which one we use.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 7.1 - 14 7.1 Greatest Common Factors; Factoring by Grouping Factoring by Grouping