Section 5.5 Factoring Trinomials, a = 1

Transcription

1 Section 5.5 Factoring Trinomials, a = 1 REVIEW Each of the following trinomials have a lead coefficient of 1. Let s see how they factor in a similar manner to those trinomials in Section 5.4. Example 1: Factor each trinomial by using the Factor Game to rewrite it as a four-term polynomial. Then use factor by grouping. a) 1x 2 + 5x 36 Key # = -36, Sum # = +5; the solution is + 9 and -4. The original trinomial: 1x 2 + 5x 36 Split the middle term into + 9x 4x: = 1x 2 + 9x 4x 36 Now show the groupings: = (1x 2 + 9x) + (- 4x 36) Now factor out the common monomial factor from each group: = x(x + 9) + - 4(x + 9) Now factor out (x + 9): = (x 4)(x + 9) We can now say that, the factors of 1x 2 + 5x 36 are (x 4) and (x + 9). b) x 2 16x + 60 This trinomial doesn t show the lead coefficient of 1; we can write it in, if we wish, for the Factor Game. 1x 2 16x + 60 Key # = -16, Sum # = +12; the solution is 6 and 6. The original trinomial: x 2 16x + 60 Split the middle term into - 10x 6x: = x 2 10x 6x + 60 Now show the groupings: = (1x 2 10x) + (- 6x + 60) Now factor out the common monomial factor from each group: = x(x 10) + - 6(x 10) Now factor out (x 10): = (x 6)(x 10) We can now say that, the factors of 1x 2 16x + 60 are (x 6) and (x 10). Factoring Trinomials, a = 1 page 5.5-1

2 Exercise 1 Factor each trinomial by using the Factor Game to rewrite it as a four-term polynomial. Then use factor by grouping. a) 1x x + 20 b) 1x 2 13x + 30 c) x 2 3x 40 d) x 2 8x + 12 Let s make a connection between the solution to the Factor Game and the binomial factors. Look back at Example 1 on the previous page: in part (a), the solutions to the Factor Game are + 9 and - 4, and the binomial factors are (x 4) and (x + 9); in part (b), the solutions to the Factor Game are - 10 and - 6, and the binomial factors are (x 6) and (x 10); In other words, the Factor Game provides the constants in the binomials. Notice also that the first term in each binomial is just x. Look at your work on this page; first check your answers for accuracy. Then, look at the solutions to the Factor Game and at the constant in each binomial. If done correctly, they should match in number and sign. Factoring Trinomials, a = 1 page 5.5-2

3 FACTORING TRINOMIALS WITH A LEAD COEFFICIENT OF 1 When the lead coefficient of a trinomial is just 1, the factoring process is much simpler (than those in Section 5.4). Factor by Grouping is no longer required. We might call these simple trinomials. What makes factoring these trinomials so much easier than those in the previous section? The answer has two parts: 1) because the first term of the trinomial is just x 2, each binomial factor can have only x as the first term, as in (x )(x ). In other words, if we see the trinomial x 2 + 7x + 12, we can start by writing the parentheses with the first term of x right away: (x )(x ). 2) the constant terms of the binomials are just the solution numbers of the Factor Game. (x + 1st solution)(x + 2nd solution) (Of course, the solutions to the Factor Game may be negative.) Example 2: Factor each trinomial using the factor game. Recognize that each has a lead coefficient of 1. a) x 2 7x + 10 b) x 2 + 2x 24 Procedure: First, identify the key and sum numbers, and play the Factor Game. Second, prepare the binomial factors by setting up the parentheses and the x s. Third, place the solutions to the Factor Game, as the constant terms in the binomials. Fourth, check this result using FOIL that it does multiply to be the trinomial. Key # = + 10 a) x 2 7x + 10 Sum # = - 7 = (x )(x ) - 5 and - 2 = (x 5)(x 2) Check: = x 2 2x 5x + 10 = x 2 7x + 10 It checks! Key # = - 24 b) x 2 + 2x 24 Sum # = + 2 = (x )(x ) + 6 and - 4 = (x + 6)(x 4) Check: = x 2 4x + 6x 24 = x 2 + 2x 24 It checks! Factoring Trinomials, a = 1 page 5.5-3

4 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). 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, is possible, using the factor game. x 2 + 5x + 30 is prime Key # = + 30 Sum # = + 5 no solution! Since the factor game has no solution, the trinomial is prime and cannot be factored. Exercise 2 Factor each trinomial. If a trinomial is not factorable, then write prime. a) x x + 20 b) x x + 30 c) x 2 + 3x 40 d) x 2 7x 30 e) x 2 + 4x 12 f) x 2 3x + 16 Factoring Trinomials, a = 1 page 5.5-4

5 g) x 2 x 30 h) x 2 13x 30 i) x 2 + 7x + 12 j) x 2 9x + 20 k) x x 24 l) x 2 9x 36 m) x x + 36 n) x 2 4x 20 o) x 2 + 8x + 16 p) x 2 10x + 25 q) x 2 12x + 36 r) x 2 + 6x + 9 Factoring Trinomials, a = 1 page 5.5-5

6 PERFECT SQUARE TRINOMIALS Look back at the last four exercises from Exercise 2, parts (o), (p), (q) and (r). Each of them factors into two binomials that are exactly the same. For example, x 2 + 8x + 16 factors into (x + 4)(x + 4). Because these factors are exactly the same, we could write the factorization as (x + 4) 2. In this way, we could say that since 49 = 7 2, it must be that 49 is a perfect square; since 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. All perfect square trinomials can be written as the square of a binomial, so the factorizations for the last four trinomials in Exercise 2 could be written as such: o) x 2 + 8x + 16 p) x 2 10x + 25 = (x + 4)(x + 4) = (x 5)(x 5) = (x + 4) 2 = (x 5) 2 q) x 2 12x + 36 r) x 2 + 6x + 9 = (x 6)(x 6) = (x + 3)(x + 3) = (x 6) 2 = (x + 3) 2 It s easy to identify perfect square trinomials. Whenever they occur (which is not that often), the solution of the Factor Game will be two factors that are exactly the same. Example 4: We already know that these trinomials are perfect square trinomials. Show that the Factor Game yields solutions that are exactly the same. a) x 2 + 8x + 16 b) x 2 10x + 25 Key # = + 16 Sum # = + 8 Key # = and + 4 Sum # = and - 5 c) x 2 12x + 36 d) x 2 + 6x + 9 Key # = + 36 Sum # = - 12 Key # = and - 6 Sum # = and + 3 Factoring Trinomials, a = 1 page 5.5-6

7 If a trinomial is a perfect square, such as x 2 10x + 25, then the Factor Game will let us know because the solutions will be exactly the same (as demonstrated in Example 4). In that case, we want to emphasize that it is a perfect square by writing the factorization as (x 5) 2. In other words, if we know that it is a perfect square, then we can show it in one step: x 2 10x + 25 = (x 5) 2 There are three indicators that will identify a trinomial as being a perfect square: i) The lead term is a perfect square, such as x 2 ; ii) the constant term that is a (positive) perfect square, as in + 25; iii) the two solutions of the Factor Game are exactly the same. Example 5: Procedure: Factor each trinomial. If the trinomial is a perfect square, write the factorization as a (binomial) 2. Recognize that the first term in each is a perfect square; also recognize that the constant term is a positive square root. Use the Factor Game to show whether or not the trinomial is a perfect square. a) x 2 + 8x + 16 Key # = + 16 b) x 2 10x + 25 Key # = + 25 Sum # = + 8 Sum # = - 10 Solutions: + 4 and + 4 Solutions: - 5 and - 5 = (x + 4) 2 (is a perfect square) = (x 5) 2 (is a perfect square) c) x 2 12x + 36 Key # = + 36 d) x 2 + 6x + 9 Key # = + 9 Sum # = - 12 Sum # = + 6 Solutions: - 6 and - 6 Solutions: + 3 and + 3 = (x 6) 2 (is a perfect square) = (x + 3) 2 (is a perfect square) e) x x + 36 Key # = + 36 f) x 2 8x + 9 Key # = + 9 Sum # = + 15 Sum # = - 8 Solutions: + 12 and + 3 Prime No solutions = (x + 12)(x + 3) (not a perfect square) Factoring Trinomials, a = 1 page 5.5-7

8 Exercise 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. a) x x + 49 b) x 2 + 9x 36 c) x 2 10x + 16 d) x x e) x 2 2x + 1 f) x 2 4x + 4 g) x x + 36 h) x x + 81 i) x x 25 j) x x 36 Factoring Trinomials, a = 1 page 5.5-8

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 Key # = +36, Sum # = +12; the solution is 6 and 6. 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. 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. Factoring Trinomials, a = 1 page 5.5-9

10 Let s take another look at factoring 4x x + 9: 4x x + 9 Key # = +36, Sum # = +12; the solution is 6 and 6. 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 6: 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 Key # = b) 16x 2 + 8x + 1 Key # = + 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 Exercise 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 5.4. If a trinomial is not factorable, then write prime. a) 9x 2 12x + 4 b) 4x 2 + 4x + 1 c) 25x x + 9 d) 81x 2 18x + 1 Factoring Trinomials, a = 1 page

11 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, Answers to each Exercise Section 5.5 Exercise 1: a) (x + 10)(x + 2) b) (x 10)(x 3) c) (x 8)(x + 5) d) (x 6)(x 2) Exercise 2: a) (x + 10)(x + 2) b) (x + 10)(x + 3) c) (x + 8)(x 5) d) (x 10)(x + 3) e) (x + 6)(x 2) f) prime g) (x 6)(x + 5) h) (x 15)(x + 2) i) (x + 3)(x + 4) j) (x 5)(x 4) k) (x 2)(x + 12) l) (x 12)(x + 3) m) (x + 4)(x + 9) n) prime o) (x + 4)(x + 4) p) (x 5)(x 5) q) (x 6)(x 6) r) (x + 3)(x + 3) Exercise 3: a) (x + 7) 2 b) (x + 12)(x 3) c) (x 8)(x 2) d) (x + 10) 2 e) (x 1) 2 f) (x 2) 2 g) (x + 4)(x + 9) h) (x + 9) 2 i) (x + 25)(x 1) j) prime Exercise 4: a) (3x 2) 2 b) (2x + 1) 2 c) (5x + 3) 2 d) (9x 1) 2 Factoring Trinomials, a = 1 page

12 Section 5.5 Focus Exercises 1. Factor each trinomial. You may do so in one step. a) x 2 12x 32 b) x 2 14x + 40 c) x x + 36 d) x 2 12x + 32 e) x 2 + 8x + 12 f) x 2 9x + 20 g) x x + 28 h) x 2 10x + 21 i) x 2 + 2x 48 j) x 2 3x 10 k) x 2 + 3x 18 l) x 2 4x 77 m) x 2 16x + 64 n) x 2 + 7x + 49 o) x 2 + 5x 50 p) x 2 x 42 Factoring Trinomials, a = 1 page

13 2. 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. a) x x + 49 b) x x c) x 2 2x + 1 d) x 2 4x + 4 e) x 2 15x + 36 f) x 2 8x + 64 g) x x 49 h) x 2 + 6x 16 i) x x + 9 j) x 2 4x 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 5.4. If a trinomial is not factorable, then write prime. a) 16x 2 + 8x + 1 b) 9x 2 + 6x + 1 c) 4x 2 20x + 25 d) 9x 2 15x + 4 e) 64x 2 16x + 1 f) 25x 2 5x + 1 Factoring Trinomials, a = 1 page