6.3 Factor Special Products *

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1 OpenStax-CNX module: m Factor Special Products * Ramon Emilio Fernandez Based on Factor Special Products 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: Factor perfect square trinomials Factor dierences of squares Factor sums and dierences of cubes Abstract Before you get started, take this readiness quiz. 1.Simplify: ( 3x ) 3. If you missed this problem, review..multiply: (m + 4). If you missed this problem, review. 3.Multiply: (x 3) (x + 3). If you missed this problem, review. We have seen that some binomials and trinomials result from special productssquaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly. 1 Factor Perfect Square Trinomials Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter. * Version 1.1: Jun 7, 017 5:30 pm

2 OpenStax-CNX module: m6450 The trinomial the square of the binomial 9x + 4x + 16 is called a perfect square trinomial. It is 3x + 4. prime factors. trinomial using the methods described in the last section, since it is of the form In this chapter, you will start with a perfect square trinomial and factor it into its You could factor this ax + bx + c. But if you recognize that the rst and last terms are squares and the trinomial ts the perfect square trinomials pattern, you will save yourself a lot of work. Here is the pattern the reverse of the binomial squares pattern. If a and b are real numbers a + ab + b = (a + b) a ab + b = (a b) (1) To make use of this pattern, you have to recognize that a given trinomial ts it. Check rst to see if the a. Next check that the last term is a perfect square, b. ab? If everything checks, you can easily write the factors. leading coe cient is a perfect square, the middle term is it the product, Example 1: How to Factor Perfect Square Trinomials Factor: 9x + 1x + 4. Solution Then check

3 OpenStax-CNX module: m Exercise Factor: (Solution on p. 1.) 4x + 1x + 9. Exercise 3 Factor: (Solution on p. 1.) 9y + 4y The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern a ab + b, which factors to (a b). The steps are summarized here. Step 1. a + ab + b Does the trinomial t the pattern? a (a) Is the rst term a perfect square? Write it as a square. (a) Is the last term a perfect square? (b) (a) Write it as a square. Check the middle term. Is it ab? Step. Write the square of the binomial. Step 3. Check by multiplying. (a) [U+198] [U+199] (b) a b (a + b) We'll work one now where the middle term is negative. Example Factor: 81y 7y Solution The rst and last terms are squares. See if the middle term ts the pattern of a trinomial. The middle term is negative, so the binomial square would be perfect square (a b). (a) [U+198] [U+ a b (

4 OpenStax-CNX module: m Are the rst and last terms perfect squares? Check the middle term. Does it match (a b)? Yes. Write as the square of a binomial. (9y 4) Check by multiplying: (9y) 9y y 7y + 16 Table 1 Exercise 5 (Solution on p. 1.) Factor: 64y 80y + 5. Exercise 6 (Solution on p. 1.) Factor: 16z 7z Remember the rst step in factoring is to look for a greatest common factor. Perfect square trinomials may have a GCF in all three terms and it should be factored out rst. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial. Example 3 Factor: 100x y 80xy + 16y. Solution

5 OpenStax-CNX module: m Is there a GCF? Yes, 4y, so factor it out. Is this a perfect square trinomial? Verify the pattern. Factor. Table Remember: Keep the factor 4y in the nal product. Check: 4y(5x ) ] 4y [(5x) 5x + 4y ( 5x 0x + 4 ) 100x y 80xy + 16y Exercise 8 (Solution on p. 1.) Factor: 8x y 4xy + 18y. Exercise 9 (Solution on p. 1.) Factor: 7p q + 90pq + 75q. Factor Dierences of Squares The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here's an example: A dierence of squares factors to a product of conjugates.

6 OpenStax-CNX module: m6450 If a and b 6 are real numbers, Remember, di erence refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted. Example 4: How to Factor a Trinomial Using the Di erence of Squares Factor: 64y 1. Solution Factor: Exercise 11 11m 1. (Solution on p. 1.)

7 OpenStax-CNX module: m Exercise 1 (Solution on p. 1.) Factor: 81y 1. Step 1. Does the binomial t the pattern? a b Is this a dierence? Are the rst and last terms perfect squares? Step. Write them as squares. (a) (b) Step 3. Write the product of conjugates. (a b) (a + b) Step 4. Check by multiplying. It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression a + b is prime! The next example shows variables in both terms. Example 5 Factor: 144x 49y. Solution 144x 49y Is this a dierence of squares? Yes. (1x) (7y) Factor as the product of conjugates. (1x 7y) (1x + 7y) Check by multiplying. (1x 7y) (1x + 7y) 144x 49y Exercise 14 (Solution on p. 1.) Factor: 196m 5n. Exercise 15 (Solution on p. 1.) Factor: 11p 9q. As always, you should look for a common factor rst whenever you have an expression to factor. Sometimes a common factor may disguise the dierence of squares and you won't recognize the perfect squares until you factor the GCF. Also, to completely factor the binomial in the next example, we'll factor a dierence of squares twice!

8 OpenStax-CNX module: m Example 6 Factor: 48x 4 y 43y. Solution 48x 4 y 43y Is there a GCF? Yes, 3y factor it out! 3y ( 16x 4 81 ) ( (4x Is the binomial a dierence of squares? Yes. 3y ) ) (9) Factor as a product of conjugates. 3y ( 4x 9 ) ( 4x + 9 ) Notice the rst binomial is also a dierence of squares! 3y ( (x) (3) ) ( 4x + 9 ) Factor it as the product of conjugates. 3y (x 3) (x + 3) ( 4x + 9 ) The last factor, the sum of squares, cannot be factored. Check by multiplying: 3y (x 3) (x + 3) ( 4x + 9 ) 3y ( 4x 9 ) ( 4x + 9 ) 3y ( 16x 4 81 ) 48x 4 y 43y Exercise 17 (Solution on p. 1.) Factor: x 4 y 3y. Exercise 18 (Solution on p. 1.) Factor: 7a 4 c 7b 4 c. 3 Key Concepts ˆ Perfect Square Trinomials Pattern: If a and b are real numbers, a + ab + b = (a + b) a ab + b = (a b) ()

9 OpenStax-CNX module: m ˆ How to factor perfect square trinomials. Step 1. Does the trinomial t the pattern? a + ab + b a ab + b Is the rst term a perfect square? (a) (a) Write it as a square. Is the last term a perfect square? (a) (b) (a) (b) Write it as a square. Check the middle term. Is it ab? (a) [U+198] a b [U+199](b) (a) [U+198] a b [U+199](b) Step. Write the square of the binomial. (a + b) (a b) Step 3. Check by multiplying. ˆ Dierence of Squares Pattern: If a, b are real numbers, ˆ How to factor dierences of squares. Step 1. Does the binomial t the pattern? a b Is this a dierence? Are the rst and last terms perfect squares? Step. Write them as squares. (a) (b) Step 3. Write the product of conjugates. (a b) (a + b) Step 4. Check by multiplying. ˆ Sum and Dierence of Cubes Pattern a 3 + b 3 = (a + b) ( a ab + b ) a 3 b 3 = (a b) ( a + ab + b ) ˆ How to factor the sum or dierence of cubes. Step a. Does the binomial t the sum or dierence of cubes pattern? Is it a sum or dierence? Are the rst and last terms perfect cubes? Step b. Write them as cubes. Step c. Use either the sum or dierence of cubes pattern. Step d. Simplify inside the parentheses Step e. Check by multiplying the factors Practice Makes Perfect Factor Perfect Square Trinomials In the following exercises, factor completely using the perfect square trinomials pattern.

10 OpenStax-CNX module: m Exercise 19 (Solution on p. 1.) 16y + 4y + 9 Exercise 0 5v + 0v + 4 Exercise 1 (Solution on p. 1.) 36s + 84s + 49 Exercise 49s + 154s + 11 Exercise 3 (Solution on p. 1.) 100x 0x + 1 Exercise 4 64z 16z + 1 Exercise 5 (Solution on p. 1.) 5n 10n Exercise 6 4p 5p Exercise 7 (Solution on p. 1.) 49x + 8xy + 4y Exercise 8 5r + 60rs + 36s Exercise 9 (Solution on p. 1.) 100y 5y + 1 Exercise 30 64m 34m + 1 Exercise 31 (Solution on p. 1.) 10jk + 80jk + 160j Exercise 3 64x y 96xy + 36y Exercise 33 (Solution on p. 1.) 75u 4 30u 3 v + 3u v Exercise 34 90p p 4 q + 50p q Factor Dierences of Squares In the following exercises, factor completely using the dierence of squares pattern, if possible. Exercise 35 (Solution on p. 1.) 5v 1 Exercise q 1 Exercise 37 (Solution on p. 1.) 4 49x Exercise s Exercise 39 (Solution on p. 1.) 6p q 54p

11 OpenStax-CNX module: m Exercise 40 98r 3 7r Exercise 41 (Solution on p. 1.) 4p + 54 Exercise 4 0b Exercise 43 (Solution on p. 1.) 11x 144y Exercise 44 49x 81y Exercise 45 (Solution on p. 13.) 169c 36d Exercise 46 36p 49q Exercise 47 (Solution on p. 13.) 16z 4 1 Exercise 48 m 4 n 4 Exercise 49 (Solution on p. 13.) 16a 4 b 3b Exercise 50 48m 4 n 43n Exercise 51 (Solution on p. 13.) x 16x + 64 y Exercise 5 p + 14p + 49 q Exercise 53 (Solution on p. 13.) a + 6a + 9 9b Exercise 54 m 6m n Exercise 55 (Solution on p. 13.) x Mixed Practice In the following exercises, factor completely. Exercise 56 (Solution on p. 13.) 64a 5 Exercise 57 11x 144 Exercise 58 (Solution on p. 13.) 7q 3 Exercise 59 4p 100

12 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 3) (x + 3) Solution to Exercise (p. 3) (3y + 4) Solution to Exercise (p. 4) (8y 5) Solution to Exercise (p. 4) (4z 9) Solution to Exercise (p. 5) y(x 3) Solution to Exercise (p. 5) 3q(3p + 5) Solution to Exercise (p. 6) (11m 1) (11m + 1) Solution to Exercise (p. 7) (9y 1) (9y + 1) Solution to Exercise (p. 7) (16m 5n) (16m + 5n) Solution to Exercise (p. 7) (11p 3q) (11p + 3q) Solution to Exercise (p. 8) y (x ) (x + ) ( x + 4 ) Solution to Exercise (p. 8) 7c (a b) (a + b) ( a + b ) (4y + 3) (6s + 7) (10x 1) (5n 1) (7x + y) (50y 1) (y 1) 10j(k + 4) 3u (5u v) (5v 1) (5v + 1) (7x ) (7x + ) 6p (q 3) (q + 3) 6 ( 4p + 9 )

13 OpenStax-CNX module: m (11x 1y) (11x + 1y) (13c 6d) (13c + 6d) (z 1) (z + 1) ( 4z + 1 ) b (3a ) (3a + ) ( 9a + 4 ) (x 8 y) (x 8 + y) (a + 3 3b) (a b) (x + 5) ( x 5x + 5 ) (8a 5) (8a + 5) 3 (3q 1) (3q + 1)