Section 7.4 Additional Factoring Techniques

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1 Section 7.4 Additional Factoring Techniques Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Factor trinomials when a = 1. Multiplying binomials (6.6) Recognize perfect square trinomials. Conjugates (6.6) Factor the difference of squares. The Factor Game (7.2) Factor by grouping (7.2) Factoring trinomials (7.3) INTRODUCTION In this section, we continue our discussion on factoring polynomials by looking at special types of trinomials and binomials. Let s first review what we know about factoring: (1) To factor means to write a polynomial in a factored form, one in which the main operation is multiplication. A factored form is the product of two polynomials. We can verify if a polynomial is in the correct factored form by multiplying the factors (often done as work off to the side or on scratch paper). For example, when we factor the trinomial 2x 2 + 7x + 6 as (2x + 3)(x + 2), we can verify that this is the correct factored form by multiplying the factors: (2x + 3)(x + 2) Distribute. = 2x 2 + 4x + 3x + 6 Combine like terms. = 2x 2 + 7x + 6 Correct! (2) Factoring is the reverse of the distributive property. In a polynomial, we can look for a common monomial factor in the terms, a monomial that may have been distributed to each term. For example, in the polynomial 2ab 3 + 2ac 4, we can see a common factor of 2a. This means that 2a may have been distributed to the terms. We can factor out, or extract, this common monomial to get a factored form: 2ab 3 + 2ac 4 = 2a(b 3 + c 4 ) Factoring Trinomials, a = 1 page 7.4-1

2 (3) For a trinomial, if the corresponding Factor Game has a winning combination, then the trinomial is guaranteed to factor. If it has no winning combination, then trinomial is not factorable. For example, the Factor Game for 2x 2 + 7x + 6 is Product = 2 6 = 12 Sum = +7 and it has a winning combination: +3 and +4. So, 2x 2 + 7x + 6 is factorable. Whereas, the Factor Game for 4x 2 + 2x + 5 is Product = 4 5 = 20 Sum = +2 has no winning combination, so 4x 2 + 2x + 5 is not factorable. Important Note: The commutative property allows the factors to be written in either order. For instance, in Example 2(b), x 2 + 2x 24 can be factored as either (x + 6)(x 4) or as (x 4)(x + 6). TRINOMIALS IN THE FORM x 2 + bx + c In Section 7.3, we discussed trinomials in the form ax 2 + bx + c, where a > 1. In this section, we explore the factoring similarities and differences of a trinomial in which a = 1: x 2 + bx + c. In this simplified form, we still use the Factor Game, and we can still use factor by grouping, as demonstrated in Example 1. After Example 1, however, we will discuss a one-step technique that works well only for factoring trinomials of the form x 2 + bx + c. (Note: this one-step technique will never work for factoring trinomials with a lad coefficient greater than 1.) Example 1: Factor each trinomial. a) x 2 + 5x 36 b) x 2 16x + 60 Procedure: Use the winning combination of the Factor Game to rewrite the trinomial as a quadrinomial. Then use factor by grouping to write it in a factored form. Factoring Trinomials, a = 1 page 7.4-2

3 Answer: a) x 2 + 5x 36 Product = 1 ( -36) = -36 and Sum = + 5. The winning combination is -4 and +9. Use the winning combination to split the middle term into 4x + 9x. = x 2 4x + 9x 36 Show the groupings. = (x 2 4x ) + ( 9x 36) Extract the common monomial factor from each group. = x(x 4) + 9(x 4) Factor out the common binomial factor, (x 4). = (x + 9)(x 4) b) x 2 16x + 60 Product = 1 (60) = 60 and Sum = -16 The winning combination is -10 and -6. Use the winning combination to split the middle term into 10x 6x. = x 2 10x 6x + 60 Show the groupings. = (x 2 10x) + ( -6x + 60) Extract the common monomial factor from each group. = x(x 10) + -6(x 10) Factor out the common binomial factor, (x 10). = (x 6)(x 10) You Try It 1 Factor each trinomial. Use Example 1 as a guide. a) x x + 20 b) x 2 13x + 30 Factoring Trinomials, a = 1 page 7.4-3

4 c) x 2 3x 40 d) x 2 8x + 12 Let s make a connection between the winning combination of the Factor Game and the binomial factors in Example 1: In part a), the winning combination is +9 and -4, and the binomial factors are (x + 9) and (x 4): x 2 + 5x 36 = (x + 9)(x 4) In part b), the winning combination is -6 and -10, and the binomial factors are (x 6) and (x 10): x 2 16x + 60 = (x 6)(x 10) Notice that, in each case, the constants of binomial factors are the exact values of the winning combination. Notice also that the first term in each binomial is just x. Next look at your work in You Try It 1. First check your answers for accuracy, then look at the winning combination of the Factor Game and at the constant in each binomial. If done correctly, they should match in number and sign. The point is this: if a trinomial of the form x 2 + bx + c is factorable there is a winning combination to its Factor Game then 1) The first term of each binomial factor is x, as in (x )(x ), and 2) the constant terms of the binomials are the factor pair in the winning combination. Factoring Trinomials, a = 1 page 7.4-4

5 Think About It 1 If a trinomial of the form x 2 + bx + c is factorable, why must the first term of each binomial be x, as in (x )(x )? Caution: This one-step technique of using the winning combination for the binomial constants works only for factoring trinomials with a lead coefficient of 1, as in x 2 + bx + c. This one-step technique will never work for a trinomial with a lead coefficient greater than 1, as in ax 2 + bx + c. Example 2 outlines the one-step technique for factoring trinomials of the form x 2 + bx + c. Example 2: Factor each trinomial using the Factor Game. a) x 2 7x + 10 b) x 2 + 2x 24 Procedure: First, identify the product and sum numbers, and play the Factor Game. Second, prepare the binomial factors by setting up the parentheses and the variables. Third, place the winning combination of the Factor Game as the constant terms in the binomials. Fourth, verify that the result is the correct factored form. Answer: a) x 2 7x + 10 Product = 1 10 = 10 and Sum = -7. The winning combination is -2 and -5. = (x )(x ) The two steps shown at left are really one step in actual work. The first step shows the start, = (x 5)(x 2) and the second step the completion. Verify: (x 5)(x 2) = x 2 2x 5x + 10 = x 2 7x + 10 Factoring Trinomials, a = 1 page 7.4-5

6 b) x 2 + 2x 24 Product = 1 (-24) = -24 and Sum = +2 = (x )(x ) The winning combination is +6 and -4. = (x + 6)(x 4) Verify: (x + 6)(x 4) = x 2 4x + 6x 24 = x 2 + 2x 24 Also, it is still possible to have a trinomial that is prime. This will happen if there are no solutions to the Factor Game. Example 3: Factor x 2 + 5x + 30, if possible. and x 2 + 5x + 30 is prime Product = 1 30 = 30 Sum = + 5 no solution! Because the factor game has no solution, the trinomial is prime cannot be factored. Because we get such quick factoring results when the leading coefficient is 1, we might call these simple trinomials. As for those trinomials with a leading coefficient greater than 1, we might call them multistep trinomials. You Try It 2 Factor each trinomial. If a trinomial is not factorable, write prime. Also, determine whether it is a simple trinomial or a multi-step trinomial. Use Examples 2 and 3 as guides. a) x x + 36 b) x 2 11x + 24 c) x 2 + 3x 40 d) 5x 2 7x 6 Factoring Trinomials, a = 1 page 7.4-6

7 e) x 2 + 4x 18 f) x 2 8x + 16 PERFECT SQUARE TRINOMIALS Consider the factoring of You Try It 2, part f): x 2 8x + 16 = (x 4)(x 4) Notice that the binomial factors are exactly the same. Because they are the same, we can write the factored form as (x 4) 2. In this way, we could say that because 49 = 7 2, it must be that 49 is a perfect square; because x 2 8x + 16 = (x 4) 2, x 2 8x + 16 is also a perfect square. x 2 8x + 16 is called a perfect square trinomial. If a trinomial is a perfect square, such as x 2 8x + 16, it is appropriate to write the factored form as the square of a binomial, (x 4) 2. Here are some examples of perfect square trinomials. a) x x + 49 b) x 2 10x + 25 = (x + 7)(x + 7) = (x 5)(x 5) = (x + 7) 2 = (x 5) 2 c) x 2 12x + 36 d) x 2 + 6x + 9 = (x 6)(x 6) = (x + 3)(x + 3) = (x 6) 2 = (x + 3) 2 It is relatively easy to identify perfect square trinomials using the Factor Game. This is because, whenever the factor pair of the winning combination is the same number, the trinomial is a perfect square trinomial, as demonstrated in Example 4. Factoring Trinomials, a = 1 page 7.4-7

8 Example 4: We already know that these trinomials are perfect square trinomials. Show that the winning combination of the Factor Game is a factor pair of the same number. a) x x + 49 b) x 2 10x + 25 c) x 2 12x + 36 d) x 2 + 6x + 9 Procedure: Play the Factor Game for each trinomial. The winning combination should be a factor pair of the same number. Product = + 49 Answer: a) Sum = + 14 Product = and + 7 b) Sum = and -5 c) Product = + 36 Sum = and -6 d) Product = + 9 Sum = and + 3 Caution: If the numbers in the factor pair of the winning combination are not the same, the trinomial is not a perfect square trinomial. You Try It 3 Factor each trinomial using the ideas developed in this section. If the trinomial is a perfect square trinomial, then write the factorization as (binomial) 2. If a trinomial is not factorable, then write prime. Use the discussion above as a guide. a) x x + 81 b) x 2 + 9x 36 c) x 2 2x + 1 d) 9x 2 12x + 4 e) x 2 10x + 16 f) x x Factoring Trinomials, a = 1 page 7.4-8

9 OTHER PERFECT SQUARE TRINOMIALS As you might imagine, we can get perfect square trinomials when the lead term is greater than 1. For example, 4x x + 9 is a perfect square trinomial. i) The lead term is a perfect square, 4x 2 = (2x) (2x); ii) the constant term is a positive perfect square, + 9; iii) the Factor Game s solutions are exactly the same; they re both + 6: 4x x + 9 Product = 4 (9) = 36 and Sum = The winning combination is +6 and +6. The numbers of the combination are the same. Here is the factorization of 4x x + 9: The original trinomial: 4x x + 9 Split the middle term into + 6x + 6x: = 4x 2 + 6x + 6x + 9 Now show the groupings: = (4x 2 + 6x) + (6x + 9) Now factor each group: = 2x(2x + 3) + 3(2x + 3) Now factor out (2x + 3): = (2x + 3)(2x + 3) = (2x + 3) 2 This is a perfect square. Here is the good news about this (or any) perfect square trinomial: if you know it s a perfect square by the Factor Game, then you can get the factorization in just one step. Look at the factorization of 4x x + 9. Notice that the square root of 4, 4, is 2 and that 9 is 3, and that those square roots show up in the factorization: it is (2x + 3) 2. Factoring Trinomials, a = 1 page 7.4-9

10 In other words, if we know that the trinomial is a perfect square (by the Factor Game), then we need only (carefully) place the numbers, the variable and the sign (plus or minus) into the outline ( ) 2. Let s take another look at factoring 4x x + 9: 4x x + 9 Product = 4 (9) = 36 and Sum = The winning combination is +6 and +6. The numbers of the combination are the same. The Factor Game guarantees that this will be a perfect square; the square root of 4 is 2, and the square root of 9 is 3; the middle term is positive, so the factorization is (2x + 3) 2. Example 5: Use the Factor Game to determine that the trinomial is a perfect square, then write its factorization as (binomial) 2. a) 25x 2 20x + 4 Product = b) 16x 2 + 8x + 1 Product = + 16 Sum = -20 Sum = + 8 Solutions: -10 and -10 Solutions: + 4 and + 4 So, it is a perfect square; So, it is a perfect square; Also, 25 = 5, 4 = 2 Also, 16 = 4, 1 = 1 So, 25x 2 20x + 4 = (5x 2) 2 So, 16x 2 + 8x + 1 = (4x + 1) 2 You Try It 4 First, determine if the trinomial is a perfect square. If it is, then write the factorization as (binomial) 2. If it not a perfect square, factor it using the method shown in Section 7.3. If a trinomial is not factorable, then write prime. Use Example 5 as a guide. a) 4x 2 12x + 9 b) 4x 2 + 4x + 1 Factoring Trinomials, a = 1 page

11 c) 25x x + 9 d) 81x 2 18x + 1 So, if the first and last terms are both perfect squares, what else is it that determines whether the whole trinomial will be a perfect square? The easiest answer is this: If the results of the Factor Game are exactly the same, then the trinomial is guaranteed to be a perfect square. This idea is summed up here: A trinomial is a perfect square if (i) (ii) both the first term and the last term are perfect squares; and the results of the Factor Game are exactly the same, THE DIFFERENCE OF SQUARES AND THE SUM OF SQUARES Let us consider two different binomials: x 2 25 and common, so we can t factor out a GCF (except 1). x In each, the terms have nothing in Notice that each term, x 2 and 25, is a perfect square: We call x 2 25 the difference of squares, and x is the sum of squares. Furthermore, we can write each binomial as a trinomial by adding in 0x as a middle term: The difference of squares, x 2 25, can be written as x 2 + 0x 25: x 2 + 0x 25 Product = 1 ( -25) = -25 and Sum = 0. = (x 5)(x + 5) The winning combination is -5 and +5. The binomials (x 5) and (x + 5) are a pair of conjugates (first discussed in Section 6.6). So, this example shows us that the difference of squares can be factored into a pair of conjugates. Factoring Trinomials, a = 1 page

12 Likewise, the sum of squares, x , can be written as x 2 + 0x + 25: x 2 + 0x + 25 Product = 1 (25) = 25 and Sum = 0. There is no winning combination. Prime! This example shows us that the sum of squares is prime, it cannot be factored. In general, The difference of squares, a 2 b 2, factors into a pair of conjugates: a 2 b 2 = (a b)(a + b) The sum of squares, c 2 + d 2, is prime. Note: 1. The binomial factors in the difference of squares can be written in either order. 2. The purpose of adding a middle term of 0x is just for explanation. We do not need to add 0x to determine the factors of either of type of these binomials. Example 6: Factor each binomial, if possible. a) y 2 4 b) x c) w 2 20 Procedure: Identify each as either the difference of squares or the sum of squares. The sum of squares is prime, but the difference of squares factors into a product of conjugates. Answer: a) (y + 2)(y 2) b) Prime. c) Prime. (20 is not a perfect square.) You Try It 5 Factor each binomial, if possible. Use Example 6 as a guide. a) x 2 49 b) x c) x d) x 2 81 d) x f) x 2 12 Factoring Trinomials, a = 1 page

13 The difference of squares can come in many forms, as long as each term is a perfect square. Some examples of perfect squares are: 49, which is (7) 2 9y 2, which is (3y) 2 25w 2, which is (5w) 2 x 4, which is ( x 2 ) 2 y 6, which is ( y 3 ) 2 and 81w 4, which is ( 9w 2 ) 2 Any of these could be a term in the difference of squares, and it will still be factorable. However, the sum of squares is prime. (There are some unusual situations when the sum of squares is not prime, but they are beyond the scope of this textbook.) Example 7: Factor each binomial, if possible. a) 9x 2 25 b) w 4 49 c) 9m 2 + 4p 2 d) 25x 4 36y 2 Procedure: Identify the terms as perfect squares, and factor accordingly. Answer: a) (3x + 5)(3x 5) 9x 2 = (3x) 2 and 25 = (5) 2 b) (w 2 7)(w 2 + 7) w 4 = ( w 2 ) 2 and 49 = (7) 2 c) Prime. 9m 2 + 4p 2 is the sum of squares. d) (5x 2 + 6y)(5x 2 6y) 25x 4 = ( 5x 2 ) 2 and 36y 2 = (6y) 2 You Try It 6 Factor each binomial, if possible. Use Example 7 as a guide. a) m 2 25p 2 b) 36w 4 1 c) 16y d) 64a 2 81b 2 e) x f) 4x 6 y 2 Factoring Trinomials, a = 1 page

14 Section 7.4, Answers to each You Try It and Think About It You Try It 1: a) (x + 10)(x + 2) b) (x 10)(x 3) c) (x 8)(x + 5) d) (x 6)(x 2) You Try It 2: a) (x + 12)(x + 3) b) (x 8)(x 3) c) (x + 8)(x 5) d) (5x + 3)(x 2) e) prime f) (x 4)(x 4) You Try It 3: a) (x + 9) 2 b) (x + 12)(x 3) c) (x 1) 2 d) (3x 2) 2 e) (x 8)(x 2) f) (x + 10) 2 You Try It 4: a) (2x 3) 2 b) (2x + 1) 2 c) (5x + 3) 2 d) (9x 1) 2 You Try It 5: a) (x + 7)(x 7) b) prime c) prime d) (x 9)(x + 9) e) prime f) prime You Try It 6: a) (m + 5p)(m 5p) b) (6w 2 1)(6w 2 + 1) c) prime d) ((8a + 9b)(8a 9b) e) prime f) (2x 3 y)(2x 3 + y) Think About It: 1. The only way to factor x 2 is x x, so the first terms of the binomials must each be x. Factoring Trinomials, a = 1 page

15 Section 7.4 Focus Exercieses Factor each trinomial, if possible. If the trinomial is a perfect square trinomial, then write its factored form as the square of a binomial. If a trinomial is not factorable, then write prime. 1. x 2 12x x 2 14x x x x 2 12x x 2 + 8x x 2 9x x 2 11x x 2 9x x 2 + 2x x 2 3x x 2 + 3x x 2 4x x 2 16x x 2 + 7x x 2 + 5x x 2 x x 2 x x 2 13x x 2 + 7x x 2 9x x x x 2 9x x x x 2 4x x 2 + 8x x 2 10x x 2 12x x 2 + 6x x x x x x x x x x x x x x 2 2x x 2 4x x 2 15x x 2 8x x x x 2 + 6x x 2 + 9x x 2 4x x 2 + 8x x 2 + 6x x 2 20x x 2 15x x 2 16x x 2 5x + 1 Factor each binomial, if possible. 49. x p y a w x 2 4 Factoring Trinomials, a = 1 page

16 55. 81a 2 25b x 2 y y a b x w c 4 d x 4 81y x a x 6 64w m 6 16p 2 Think Outside the Box. Each of these can be factored more than once. In other words, more than one technique (or even the same technique) can be applied in different steps of the factoring process. 67. x 3 7x 2 8x 68. 5x 2 15x x 3 27x 70. y x 3 12x x x 2 + 4x + 10 Factoring Trinomials, a = 1 page