6.1 Greatest Common Factor and Factor by Grouping *
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1 OpenStax-CNX module: m Greatest Common Factor and Factor by Grouping * Ramon Emilio Fernandez Based on Greatest Common Factor and Factor by Grouping by OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of this section, you will be able to: Abstract Find the greatest common factor of two or more expressions Factor the greatest common factor from a polynomial Factor by grouping Before you get started, take this readiness quiz. 1.Factor 56 into primes. If you missed this problem, review. 2.Find the least common multiple (LCM) of 18 and 24. If you missed this problem, review. 3.Multiply: 3a (7a + 8b). If you missed this problem, review. 1 Find the Greatest Common Factor of Two or More Expressions Earlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. * Version 1.1: Jun 27, :29 pm
2 OpenStax-CNX module: m We have learned how to factor numbers to nd the least common multiple (LCM) of two or more numbers. Now we will factor expressions and nd the greatest common factor of two or more expressions. The method we use is similar to what we used to nd the LCM. The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. We summarize the steps we use to nd the greatest common factor. Step 1.Factor each coecient into primes. Write all variables with exponents in expanded form. Step 2.List all factorsmatching common factors in a column. In each column, circle the common factors. Step 3.Bring down the common factors that all expressions share. Step 4.Multiply the factors. The next example will show us the steps to nd the greatest common factor of three expressions. Example 1 Find the greatest common factor of 21x 3, 9x 2, 15x. Solution
3 OpenStax-CNX module: m Factor each coecient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors. Table 1 The GCF of 21x 3, 9x 2 and 15x is 3x. Exercise 2 (Solution on p. 13.) Find the greatest common factor: 25m 4, 35m 3, 20m 2. Exercise 3 (Solution on p. 13.) Find the greatest common factor: 14x 3, 70x 2, 105x. 2 Factor the Greatest Common Factor from a Polynomial It is sometimes useful to represent a number as a product of factors, for example, 12 as 2 6 or 3 4. In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as 3x 2 +15x, and end with its factors, 3x (x + 5). To do this we apply the Distributive Property in reverse. We state the Distributive Property here just as you saw it in earlier chapters and in reverse. If a, b, and c are real numbers, then a (b + c) = ab + ac and ab + ac = a (b + c) (1) The form on the left is used to multiply. The form on the right is used to factor. So how do you use the Distributive Property to factor a polynomial? You just nd the GCF of all the terms and write the polynomial as a product! Example 2: How to Use the Distributive Property to factor a polynomial Factor: 8m 3 12m 2 n + 20mn 2.
4 OpenStax-CNX module: m Solution Factor: Exercise 5 (Solution on p. 13.) 9xy 2 + 6x2 y y 3. Exercise 6 Factor: 3 (Solution on p. 13.) 2 3 3p 6p q + 9pq. Step 1.Find the GCF of all the terms of the polynomial. Step 2.Rewrite each term as a product using the GCF. Step 3.Use the reverse Distributive Property to factor the expression. Step 4.Check by multiplying the factors. We use factor as both a noun and a verb: Noun: Verb: 7 is a factor of 14 factor 3 from 3a + 3 (2)
5 OpenStax-CNX module: m Exercise 7 (Solution on p. 13.) Factor: 2x x 2. Exercise 8 (Solution on p. 13.) Factor: 6y 3 15y 2. Example 3 Factor: 8x 3 y 10x 2 y xy 3. Solution The GCF of 8x 3 y, 10x 2 y 2, and 12xy 3 is 2xy. Rewrite each term using the GCF, 2xy. Factor the GCF. Check: 2xy ( 4x 2 5xy + 6y 2) 2xy 4x 2 2xy 5xy + 2xy 6y 2 8x 3 y 10x 2 y xy 3 Table 2
6 OpenStax-CNX module: m Exercise 10 (Solution on p. 13.) Factor: 15x 3 y 3x 2 y 2 + 6xy 3. Exercise 11 (Solution on p. 13.) Factor: 8a 3 b + 2a 2 b 2 6ab 3. When the leading coecient is negative, we factor the negative out as part of the GCF. Exercise 12 (Solution on p. 13.) Factor: 7a a 2 14a. So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial. Example 4 Factor: 3y (y + 7) 4 (y + 7). Solution The GCF is the binomial y + 7. Factor the GCF, (y + 7). Check on your own by multiplying. Table 3 Exercise 14 (Solution on p. 13.) Factor: 4m (m + 3) 7 (m + 3). Exercise 15 (Solution on p. 13.) Factor: 8n (n 4) + 5 (n 4).
7 OpenStax-CNX module: m Factor by Grouping Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will nd a common factor emerges from both parts. Not all polynomials prime, some polynomials are prime. Example 5: How to Factor a Polynomial by Grouping can be factored. Just like some numbers are Factor by grouping: Solution xy + 3y + 2x + 6. Exercise 17 Factor by grouping: (Solution on p. 13.) xy + 8y + 3x Exercise 18 Factor by grouping: (Solution on p. 13.) ab + 7b + 8a + 56.
8 OpenStax-CNX module: m Step 1.Group terms with common factors. Step 2.Factor out the common factor in each group. Step 3.Factor the common factor from the expression. Step 4.Check by multiplying the factors. Example 6 Factor by grouping: ax 2 + 3x 2x 6b6x 2 3x 4x + 2. Solution a There is no GCF in all four terms. x 2 + 3x 2x 6 Separate into two parts. x 2 + 3x 2x 6 Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms. x (x + 3) 2 (x + 3) Factor out the common factor. (x + 3) (x 2) Check on your own by multiplying. Exercise 20 (Solution on p. 13.) Factor by grouping: ax 2 + 2x 5x 10b20x 2 16x 15x Key Concepts ˆ How to nd the greatest common factor (GCF) of two expressions. Step a. Factor each coecient into primes. Write all variables with exponents in expanded form. Step b. List all factorsmatching common factors in a column. In each column, circle the common factors. Step c. Bring down the common factors that all expressions share. Step d. Multiply the factors. ˆ Distributive Property: If a, b, and c are real numbers, then a (b + c) = ab + ac and ab + ac = a (b + c) (3) The form on the left is used to multiply. The form on the right is used to factor. ˆ How to factor the greatest common factor from a polynomial. Step a. Find the GCF of all the terms of the polynomial. Step b. Rewrite each term as a product using the GCF. Step c. Use the reverse Distributive Property to factor the expression. Step d. Check by multiplying the factors.
9 OpenStax-CNX module: m ˆ Factor as a Noun and a Verb: We use factor as both a noun and a verb. ˆ How to factor by grouping. Noun: 7 is a factor of 14 Verb: factor 3 from 3a + 3 Step a. Group terms with common factors. Step b. Factor out the common factor in each group. Step c. Factor the common factor from the expression. Step d. Check by multiplying the factors. (4) Practice Makes Perfect Find the Greatest Common Factor of Two or More Expressions In the following exercises, nd the greatest common factor. Exercise 21 (Solution on p. 13.) 10p 3 q, 12pq 2 Exercise 22 8a 2 b 3, 10ab 2 Exercise 23 (Solution on p. 13.) 12m 2 n 3, 30m 5 n 3 Exercise 24 28x 2 y 4, 42x 4 y 4 Exercise 25 (Solution on p. 13.) 10a 3, 12a 2, 14a Exercise 26 20y 3, 28y 2, 40y Exercise 27 (Solution on p. 13.) 35x 3 y 2, 10x 4 y, 5x 5 y 3 Exercise 28 27p 2 q 3, 45p 3 q 4, 9p 4 q 3 Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. Exercise 29 (Solution on p. 13.) 6m + 9 Exercise 30 14p + 35 Exercise 31 (Solution on p. 13.) 9n 63 Exercise 32 45b 18 Exercise 33 (Solution on p. 13.) 3x 2 + 6x 9
10 OpenStax-CNX module: m Exercise 34 4y 2 + 8y 4 Exercise 35 (Solution on p. 13.) 8p 2 + 4p + 2 Exercise 36 10q q + 20 Exercise 37 (Solution on p. 13.) 8y y 2 Exercise 38 12x 3 10x Exercise 39 (Solution on p. 13.) 5x 3 15x x Exercise 40 8m 2 40m + 16 Exercise 41 (Solution on p. 13.) 24x 3 12x x Exercise 42 24y 3 18y 2 30y Exercise 43 (Solution on p. 14.) 12xy x 2 y 2 30y 3 Exercise 44 21pq p 2 q 2 28q 3 Exercise 45 (Solution on p. 14.) 20x 3 y 4x 2 y xy 3 Exercise 46 24a 3 b + 6a 2 b 2 18ab 3 Exercise 47 (Solution on p. 14.) 2x 4 Exercise 48 3b + 12 Exercise 49 (Solution on p. 14.) 2x x 2 8x Exercise 50 5y y 2 15y Exercise 51 (Solution on p. 14.) 4p 3 q 12p 2 q pq 2 Exercise 52 6a 3 b 12a 2 b ab 2 Exercise 53 (Solution on p. 14.) 5x (x + 1) + 3 (x + 1) Exercise 54 2x (x 1) + 9 (x 1) Exercise 55 (Solution on p. 14.) 3b (b 2) 13 (b 2)
11 OpenStax-CNX module: m Exercise 56 6m (m 5) 7 (m 5) Factor by Grouping In the following exercises, factor by grouping. Exercise 57 (Solution on p. 14.) ab + 5a + 3b + 15 Exercise 58 cd + 6c + 4d + 24 Exercise 59 (Solution on p. 14.) 8y 2 + y + 40y + 5 Exercise 60 6y 2 + 7y + 24y + 28 Exercise 61 (Solution on p. 14.) uv 9u + 2v 18 Exercise 62 pq 10p + 8q 80 Exercise 63 (Solution on p. 14.) u 2 u + 6u 6 Exercise 64 x 2 x + 4x 4 Exercise 65 (Solution on p. 14.) 9p 2 3p 20 Exercise 66 16q 2 8q 35 Exercise 67 (Solution on p. 14.) mn 6m 4n + 24 Exercise 68 r 2 3r r + 3 Exercise 69 (Solution on p. 14.) 2x 2 14x 5x + 35 Exercise 70 4x 2 36x 3x + 27 Mixed Practice In the following exercises, factor. Exercise 71 (Solution on p. 14.) 18xy 2 27x 2 y Exercise 72 4x 3 y 5 x 2 y xy 4 Exercise 73 (Solution on p. 14.) 3x 3 7x 2 + 6x 14 Exercise 74 x 3 + x 2 x 1 Exercise 75 (Solution on p. 14.) x 2 + xy + 5x + 5y
12 OpenStax-CNX module: m Exercise 76 5x 3 3x 2 + 5x 3
13 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 3) 5m 2 Solution to Exercise (p. 3) 7x Solution to Exercise (p. 4) 3y 2 ( 3x + 2x 2 + 7y ) Solution to Exercise (p. 4) 3p ( p 2 2pq + 3q 2) Solution to Exercise (p. 5) 2x 2 (x + 6) Solution to Exercise (p. 5) 3y 2 (2y 5) Solution to Exercise (p. 6) 3xy ( 5x 2 xy + 2y 2) Solution to Exercise (p. 6) 2ab ( 4a 2 + ab 3b 2) Solution to Exercise (p. 6) 7a ( a 2 3a + 2 ) Solution to Exercise (p. 6) (m + 3) (4m 7) Solution to Exercise (p. 6) (n 4) (8n + 5) Solution to Exercise (p. 7) (x + 8) (y + 3) Solution to Exercise (p. 7) (a + 7) (b + 8) Solution to Exercise (p. 8) a (x 5) (x + 2) b (5x 4) (4x 3) Solution to Exercise (p. 9) 2pq Solution to Exercise (p. 9) 6m 2 n 3 Solution to Exercise (p. 9) 2a Solution to Exercise (p. 9) 5x 3 y Solution to Exercise (p. 9) 3 (2m + 3) Solution to Exercise (p. 9) 9 (n 7) Solution to Exercise (p. 9) 3 ( x 2 + 2x 3 ) 2 ( p 2 + 4p + 1 ) 8y 2 (y + 2) 5x ( x 2 3x + 4 )
14 OpenStax-CNX module: m x ( 8x 2 4x + 5 ) 6y 2 ( 2x + 3x 2 5y ) 4xy ( 5x 2 xy + 3y 2) 2 (x + 4) 2x ( x 2 9x + 4 ) 4pq ( p 2 + 3pq 4q ) (x + 1) (5x + 3) (b 2) (3b 13) (b + 5) (a + 3) (y + 5) (8y + 1) (u + 2) (v 9) (u 1) (u + 6) (3p 5) (3p + 4) (n 6) (m 4) (x 7) (2x 5) 9xy (3x + 2y) ( x ) (3x 7) (x + y) (x + 5) Glossary Denition 4: factoring Splitting a product into factors is called factoring. Denition 4: greatest common factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
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