Copyright © 2016, 2012, 2008 Pearson Education, Inc. 1 Factoring out the Greatest Common Factor

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 2 The Greatest Common Factor; Factoring by Grouping 1. Find the greatest common factor of a list of terms. 2. Factor out the greatest common factor. 3. Factor by grouping.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 3 Factoring To factor a number means to write it as a product of two or more numbers. The product is a factored form of the number. Consider an example. Factoring is a process that “undoes” multiplying. We multiply 6 · 2 to obtain 12, but we factor 12 by writing it as 6 · 2. Other factored forms of 12 are Factors

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 4 Objective 1 Find the greatest common factor of a list of terms.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 5 Greatest Common Factor An integer that is a factor of two or more integers is a common factor of those integers. For example, 6 is a common factor of 18 and 24, because 6 is a factor of both 18 and 24. Other common factors of 18 and 24 are 1, 2, and 3. The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. Thus, 6 is the greatest common factor of 18 and 24, because it is the largest of their common factors.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 6 Finding the Greatest Common Factor (GCF) Step 1 Factor. Write each number in prime factored form. Step 2 List common factors. List each prime number or each variable that is a factor of every term in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.) Step 3 Choose least exponents. Use as exponents on the common prime factors the least exponents from the prime factored forms. Step 4 Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 7 Divisibility Tests A Whole Number Divisible by Must Have the Following Property: 2Ends in 0, 2, 4, 6, or 8 3Sum of its digits is divisible by 3. 4Last two digits form a number divisible by 4 5Ends in 0 or 5

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 8 A Whole Number Divisible by Must Have the Following Property: 6Divisible by both 2 and 3 8Last three digits form a number divisible by 8 9Sum of its digits is divisible by 9. 10Ends in 0 Divisibility Tests

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 9 Find the greatest common factor for each list of numbers. a. 50, = 2 · 5 · 5 75 = 3 · 5 · 5 b. 12, 18, 26, = 2 · 2 · 3 18 = 2 · 3 · 3 26 = 2 · = 2 · 2 · 2 · 2 · 2 Example 1 Finding the Greatest Common Factor (Numbers)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 10 Find the greatest common factor for each list of numbers. c. 22, 23, = 2 · = 1 · = 2 · 2 · 2 · 3 There are no primes common to all three numbers, so the GCF is 1. Example 1 Finding the Greatest Common Factor (Numbers) (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 11 Finding the Greatest Common Factor (GCF) The greatest common factor can also be found for a list of variable terms. The exponent on a variable in the GCF is the least exponent that appears in all the common factors.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 12 Find the greatest common factor for each list of terms. a. 16r 9, 10r 15, 8r 12 16r 9 = 2 · 2 · 2 · 2 · r 9 10r 15 = 2 · 5 · r 15 8r 12 = 2 · 2 · 2 · r 12 The greatest common factor of the coefficients is 2. The least exponent on r is 9. Example 2 Finding the Greatest Common Factor (Variable Terms)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 13 Find the greatest common factor for each list of terms. b. s 4 t 5, s 3 t 6, s 9 t 2 There is an s in each term, and 3 is the least exponent. There is a t in each term, and 2 is the least exponent. Example 2 Finding the Greatest Common Factor (Variable Terms) (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 14 Objective 2 Factor out the greatest common factor.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 15 Factor out the greatest common factor. Factoring a polynomial is the process of writing a polynomial sum in factored form as a product. For example, the polynomial 3m + 12 has two terms, 3m and 12. The greatest common factor of these two terms is 3. We can write 3m + 12 so that each term is a product with 3 as one factor. The factored form of 3m + 12 is 3(m + 4). This process is called factoring out the greatest common factor.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 16 Write in factored form by factoring out the greatest common factor. a.6x x The GCF is 6x 2. Example 3 Factoring Out the Greatest Common Factor

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 17 Write in factored form by factoring out the greatest common factor. b. The GCF is 5t 4 c. The GCF is r 10 Example 3 Factoring Out the Greatest Common Factor (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 18 Write in factored form by factoring out the greatest common factor. d. The GCF is 4p 4 q 2. Example 3 Factoring Out the Greatest Common Factor (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 19 Write –2x 5 – 8x 4 + 4x 3 in factored form by factoring out a negative common factor. Factor out –2x 3. Example 4 Factoring Out a Negative Common Factor

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 20 Write in factored form by factoring out the greatest common factor. a. b. Example 5 Factoring Out the Greatest Common Factor

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 21 Objective 3 Factoring by grouping.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 22 Factoring by grouping. When a polynomial has four terms, common factors can sometimes be used to factor by grouping.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 23 Factor by grouping. a. pq + 5q + 2p + 10 Group the first two terms and the last two terms. You can check by multiplying. Example 6 Factoring by Grouping

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 24 Factor by grouping. b. c. Example 6 Factoring by Grouping (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 25 Factor by grouping. d. Example 6 Factoring by Grouping (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 26 Factoring a Polynomial with Four Terms by Grouping Step 1 Group the terms. Collect the terms into two groups so that each group has a common factor. Step 2 Factor within the groups. Factor out the greatest common factor from each group. Step 3 If possible, factor the entire polynomial. Factor out a common binomial factor from the results of Step 2. Step 4 If necessary, rearrange terms. If Step 2 does not result in a common binomial factor, try a different grouping. Always check the factored form by multiplying.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 27 Factor by grouping. a. This does not lead to a common factor so we try rearranging the terms. Example 7 Rearranging Terms before Factoring by Grouping

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 28 Factor by grouping. b. Rearrange the terms to obtain two groups that each have a common factor. Example 7 Rearranging Terms before Factoring by Grouping (cont.)