1. Section 3.1 Homework Questions?

2. Section Concepts 3.1 Greatest Common Factor and Factoring by Grouping Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Identifying the Greatest Common Factor 2.Factoring out the Greatest Common Factor 3.Factoring out a Negative Factor 4.Factoring out a Binomial Factor 5.Factoring by Grouping

3. Section 3.1 Greatest Common Factor and Factoring by Grouping 1.Identifying the Greatest Common Factor (continued) Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To factor an integer means to write the integer as a product of two or more integers. To factor a polynomial means to express the polynomial as a product of two or more polynomials.

4. Section 3.1 Greatest Common Factor and Factoring by Grouping 1.Identifying the Greatest Common Factor (continued) Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We begin our study of factoring by factoring integers. The number 20, for example, can be factored as The product consists only of prime numbers and is called the prime factorization.

5. Section 3.1 Greatest Common Factor and Factoring by Grouping 1.Identifying the Greatest Common Factor Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The greatest common factor (denoted GCF) of two or more integers is the greatest factor common to each integer.

6. Example 1Identifying the GCF Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the greatest common factor. a. 24 and 36 b. 40 and 60c. 12, 24, and 30

7. TIP: Slide 7 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notice that the expressions and share a common numerical factor of 5, and common variable factors of a, and b. The GCF is the product of the largest common numerical factor (5) and the common variable factors, where each variable is raised to the lowest power to which it occurs in all the original expressions. The GCF of and is

8. Example 2Identifying the Greatest Common Factor Slide 8 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the GCF among each group of terms. a. b.c.

9. Example Solution: 3Identifying the Greatest Common Factor Slide 9 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the GCF among each group of terms. a.,, b.,,

10. Example 3Finding the Greatest Common Binomial Factor Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the greatest common factor between the terms:

11. Example Solution: 3Finding the Greatest Common Binomial Factor Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The only common factor is the binomial The GCF is The Greatest Common Factors in Examples 1 and 2 were monomials. The GCF In Example 3 was a binomial. Common factors which are binomials occur frequently when we “factor by grouping” a skill we will learn later in this presentation.

12. Section 3.1 Greatest Common Factor and Factoring by Grouping 2.Factoring out the Greatest Common Factor (continued) Slide 12 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The process of factoring a polynomial is the reverse process of multiplying polynomials. To factor out the greatest common factor means to express the original polynomial as the product of the GCF and some new factor (polynomial).

13. Section 3.1 Greatest Common Factor and Factoring by Grouping 2.Factoring out the Greatest Common Factor Slide 13 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multiply/Distribute Factor out the greatest common factor

14. PROCEDUREFactoring out the Greatest Common Factor Slide 14 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Identify the GCF of all terms of the polynomial. Step 2 Write the original polynomial as the product of the GCF and another factor. Note: To check the factorization, multiply the polynomials to remove parentheses.

15. Example 4Factoring out the Greatest Common Factor Slide 15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor out the GCF. a.b.

16. Example 5Factoring out the Greatest Common Factor Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor out the GCF. a.b.

17. Section 3.1 Greatest Common Factor and Factoring by Grouping 2.Factoring out the Greatest Common Factor Slide 17 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The greatest common factor of the polynomial If we factor out the GCF, we have 1 A polynomial whose only factors are itself and 1 is called a prime polynomial. A prime polynomial cannot be factored further.

18. Avoiding Mistakes Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In Example 5(a), the GCF is 16x, and the factorization of is. If the Greatest Common Factor has been factored out the second factor must be prime ( have no other factors except one). If in Example 5(a) if a GCF of 8x were used the factorization would have been. The second factor is not prime as it contains a common factor of 2. When factoring out the Greatest Common Factor remember to check to make sure the second factor is prime to ensure you have removed the GCF.

19. Section 3.1 Greatest Common Factor and Factoring by Grouping 3.Factoring out a Negative Factor Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Usually it is advantageous to factor out the opposite of the GCF when the leading coefficient of the polynomial is negative. Notice that this changes the signs of the remaining terms inside the parentheses.

20. Example 6Factoring out a Negative Factor Slide 20 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor outfrom the polynomial

21. Example 7Factoring out a Negative Factor Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor out the quantity from the polynomial

22. Section 3.1 Greatest Common Factor and Factoring by Grouping 4.Factoring out a Binomial Factor Slide 22 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The same process which was used to factor out a common factor which was a monomial can also be used to factor out a common factor that consists of more than one term.

23. Example 8Factoring out a Binomial Factor Slide 23 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor out the GCF:

24. Section 3.1 Greatest Common Factor and Factoring by Grouping 5.Factoring by Grouping Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Given a four-term polynomial, we will factor it as a product of two binomials. The process is called factoring by grouping.

25. PROCEDUREFactoring by Grouping Slide 25 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To factor a four-term polynomial by grouping: Step 1 Identify and factor out the GCF from all four terms. (Should one exist) Step 2 Factor out the GCF from the first pair of terms. Factor out the GCF from the second pair of terms. (Sometimes it is necessary to factor out the opposite of the GCF.) Step 3 If the two terms share a common binomial factor, factor out the binomial factor.

26. Example 9Factoring by Grouping Slide 26 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor by grouping:

27. Example Solution: 10Factoring by Grouping (continued) Slide 27 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor by grouping

28. Example Solution: 11Factoring by Grouping (continued) Slide 28 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor by grouping

29. TIP: Slide 29 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. One frequently asked question when factoring is whether the order can be switched between the factors. The answer is yes. Because multiplication is commutative, the order in which the factors are written does not matter.

30. Example 12Factoring by Grouping Slide 30 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Factor by grouping.

31. Section 3.1 Greatest Common Factor and Factoring by Grouping You Try a.a. b., Identify the GCF Factor out the GCF

32. Section Greatest Common Factor and Factoring by Grouping You Try a.a. b. 3.1 Factor

33. Section Greatest Common Factor and Factoring by Grouping You Try a. b. 3.1 Factor by Grouping