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Factoring Polynomials and Solving Equations by Factoring Copyright © Cengage Learning. All rights reserved. 5 Section 5.4 Factoring Trinomials with a Leading Coefficient Other Than 1 Copyright © Cengage Learning. All rights reserved. Objectives 1 Factor a trinomial of the form ax2 + bx + c using trial and error. 2 Factor a trinomial of the form ax2 + bx + c by grouping (ac method). 3 Factor a polynomial involving a perfect-square trinomial. 3 Factoring Trinomials with a Leading Coefficient Other Than 1 We saw how to factor trinomials whose leading coefficients are 1. We now show how to factor trinomials whose leading coefficients are other than 1. 4 1. Factor a trinomial of the form ax2 + bx + c using trial and error 5 Factor a trinomial of the form ax2 + bx + c using trial and error We must consider more combinations of factors when we factor trinomials with leading coefficients other than 1. Factoring General Trinomials Using Trial and Error 1. Write the trinomial in descending powers of one variable. 2. Factor out any GCF (including –1 if that is necessary to make the coefficient of the first term positive). 6 Factor a trinomial of the form ax2 + bx + c using trial and error 3. If the sign of the third term is +, the signs between the terms of the binomial factors are the same as the sign of the middle term. If the sign of the third term is –, the signs between the terms of the binomial factors are opposites. 4. Try combinations of first terms and last terms until you find one that works, or until you exhaust all the possibilities. If no combination works, the trinomial is prime. 5. Check the factorization by multiplication. 7 Example Factor: 2x2y – 8x3 + 3xy2. Solution: Step 1: Write the trinomial in descending powers of x. –8x3 + 2x2y + 3xy2 Step 2: Factor out the negative of the GCF, which is –x. –8x3 + 2x2y + 3xy2 = –x(8x2 – 2xy – 3y2) Step 3: Because the sign of the third term of the trinomial factor is –, the signs within its binomial factors will be opposites. 8 Example – Solution cont’d Step 4: Find the binomial factors of the trinomial. –8x3 + 2x2y + 3xy2 = –x(8x2 – 2xy – 3y2) = –x(2x + y)(4x – 3y) Step 5: Check by multiplication. –x(2x + y)(4x – 3y) = –x(8x2 – 6xy + 4xy – 3y2) = –x(8x2 – 2xy – 3y2) = –8x3 + 2x2y + 3xy2 = 2x2y – 8x3 + 3xy2 9 2. Factor a trinomial of the form ax2 + bx + c by grouping (ac method) 10 Factor a trinomial of the form ax2 + bx + c by grouping (ac method) Another way to factor trinomials of the form ax2 + bx + c uses the grouping (ac method). For example, to factor 6x2 – 17x + 5 (Example 2) by grouping, we note that a = 6, b = –17, and c = 5 and proceed as follows: 1. Determine the product ac: 6(+5) = 30. This is the key number. 2. Find two factors of the key number 30 whose sum is –17. Two such factors are –15 and –2. –15(–2) = 30 and –15 + (–2) = –17 11 Factor a trinomial of the form ax2 + bx + c by grouping (ac method) 3. Use –15 and –2 as coefficients of two terms to be placed between 6x2 and 5 to replace –17x. 6x2 – 17x + 5 = 6x2 – 15x – 2x + 5 4. Factor the right side of the previous equation by grouping. 6x2 – 15x – 2x + 5 = 3x(2x – 5) – 1(2x – 5) = (2x – 5)(3x – 1) Factor the GCF, 3x, from (6x2 – 15x) and –1 from (–2x + 5). Factor out the GCF, (2x – 5). Verify this factorization by multiplication. 12 Example Factor 4y2 + 12y + 5 by grouping. Solution: To factor this trinomial by grouping, we note that it is written in the form ay2 + by + c, with a = 4, b = 12, and c = 5. Since a = 4 and c = 5, we have ac = 20. We now find two factors of 20 whose sum is 12. Two such factors are 10 and 2. 13 Example – Solution cont’d We use these factors as coefficients of two terms to be placed between 4y2 and 5 to replace +12y. 4y2 + 12y + 5 = 4y2 + 10y + 2y + 5 Finally, we factor the right side of the previous equation by grouping. Factor the GCF, 2y, 4y2 + 10y + 2y + 5 = 2y(2y + 5) + (2y + 5) from (4y2 + 10y). = 2y(2y + 5) + 1 (2y + 5) = (2y + 5)(2y + 1) Check by multiplication. Write the coefficient of 1. Factor out the GCF, (2y + 5). 14 3. Factor a polynomial involving a perfect-square trinomial 15 Example Factor: 4x2 – 20x + 25. Solution: 4x2 – 20x + 25 is a perfect-square trinomial, because • The first term is 4x2 is the square of 2x: (2x)2 = 4x2. • The last term 25 is the square of 5: 52 = 25. • The middle term –20x is the negative of twice the product of 2x and 5. 16 Example – Solution cont’d Thus, 4x2 – 20x + 25 = (2x)2 – 2(2x)(5) + 52 = (2x – 5)2 Check by multiplication. 17