1. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 Factoring: Use of the Distributive Property Example 1: Find the GCF for 36 and 60. Objective A: Finding the greatest common factor The greatest common factor is the largest number that divides evenly into a set of numbers. For example, the GCF of 12 and 18 would be 6 because 6 is the largest number that divides evenly into both numbers. Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common. Your Turn Problem #1 Find the GCF for 28 and 70.

2. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 2 Example 2: Find the GCF for x 3 and x 5. Objective A: Finding the greatest common factor Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common. Your Turn Problem #2 Find the GCF for x 8 and x 12. So, when finding the GCF if variable terms, use the variable with the lowest exponent.

3. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 3 Example 3: Find the GCF for 84x 7 and 120x 3. Objective A: Finding the greatest common factor Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common then take the variable with the lowest exponent. Your Turn Problem #3 Find the GCF for 42x 5 and 56x 11.

4. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 4 Example 4: Find the GCF for (x + 5)(x - 3) and (x - 7)(x - 3). Objective A: Finding the greatest common factor Step 1. Write the product of each. Step 2. Circle the factors they have in common. In this case, the common factor is a binomial. (x + 5)(x - 3) (x - 7)(x - 3). Your Turn Problem #4 Find the GCF for (x + a)(a - b) and (x + b)(a - b). Answer: a - b

5. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 5 Example 5: Find the GCF for x 2 – 5x – 36 and x 2 +x –12 Objective A: Finding the greatest common factor Step 1. Write factored form of each. Step 2. Circle the factors they have in common. In this case, the common factor is a binomial. Answer: x - 4 Your Turn Problem #5

6. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 6 Objective B: Factoring a Monomial from a PolynomialGeneral Statement ab + ac = a(b + c) The process of finding a common monomial factor is the Distributive Property in reverse. Procedure: To factor a monomial from a polynomial Step 1. Find the greatest common factor (GCF) of all terms of the polynomial. Step 2. Divide each term by this GCF. Step 3.Write the answer in the form: (GCF)(quotients of each term). Note: Steps 2 and 3 are the Distributive Property worked backwards. 1. Find the greatest common factor of these terms. The greatest common factor of each term is 6. Notes:1. The greatest common factor of two or more integers is the greatest integer that is a common factor of all the integers. 2.The greatest common factor of variable factors is the smallest exponent of each variable that is common to all. 2. Divide each term by the GCF. 3. Write the answer: (GCF)(quotients of each term) 6(3x + 4) Your Turn Problem #6

7. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 7 1. Find the GCF for each term. GCF = 6a 3 b 2 2. Divide each term by the GCF. Your Turn Problem #7 3.Write the answer: (GCF)(quotients of each term)

8. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 8 Objective C: Factoring a Binomial from a Polynomial (The greatest common factor is a binomial) General Statement a(x+y) +b(x+y) = (x+y)(a+b) GCF = (3a + 2b). 2x – 5y Your Turn Problem #8 1. Find the GCF for each term. 2. Divide each term by the GCF. 3.Write the answer: (GCF)(quotients of each term)

9. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 9 Objective D: Factoring by Grouping Some polynomials can be factored by grouping terms in such a way that a common binomial factor is found. Example: ax + bx + ay + by 1 st, Factor the GCF from the first two terms and the last two terms. x(a+b)+ y(a+b) 2 nd, Factor the common binomial from the expression. Answer: (a+b)(x+y) Notes: The goal is to obtain a common binomial in both terms. Sometimes the order of the polynomial may have to be rearranged to achieve the desired outcome. If the first term of the second pair is negative, factor out the negative along with the GCF. Factor out a 3x from the first pair. Since the first term of the second pair is negative, factor out a –2y. Lastly, factor out the common binomial. Your Turn Problem #9

10. 4.1 GCF and Factoring by Grouping BobsMathClass.Com Copyright © 2010 All Rights Reserved. 10 Since the first two two terms do not have a common factor, we will need to rearrange the terms to factor by grouping. Answer: Your Turn Problem #10 The End. B.R. 1-27-09