Factoring Trinomials Factoring ax 2 + bx + c Step 1Find pairs whose product is c. Find all pairs of integers whose product is c, the third term of the trinomial. Step 2Find pairs whose sum is b. Choose the pair whose sum is b, the coefficient of the middle term. If there are no such integers, the polynomials cannot be factored. A polynomial that cannot be factored with integer coefficient is a prime polynomial.

Slide We can use an X to help us in factoring ax 2 +bx+c, when a=1 What we need it to multiply to c b What we need it to add to, Factor

Slide EXAMPLE 1 Factor. a 2 + 9a + 20 Step 1 Find pairs of numbers whose product is 20. Step 2 Write sums of those numbers

Slide continued The coefficient of the middle term is 9, so the required numbers are 5 and 4. The factored form is a 2 + 9a + 20 = (a + 5)(a + 4). Check (a + 5)(a + 4) = a 2 + 9a + 20

Slide EXAMPLE 2 Factor t 2 + 3t – 5. Look for two expressions whose product is –5 and whose sum is 3. There are no such quantities. Therefore, the trinomial cannot be factored and is prime.

Slide EXAMPLE 3 Factor p 2 – 5pq + 6q 2. Look for two expressions whose product is 6q 2 and whose sum is –5q. The quantities –3q and –2q have the necessary product and sum, so p 2 – 5pq + 6q 2 = (p – 3q)(p – 2q).

Slide EXAMPLE 4 Factor 3a a 2 – 15a. Start by factoring out the GCF, 3a. 3a a 2 – 15a = 3a(a 2 + 4a – 5) To factor a 2 + 4a – 5, look for two integers whose product is –5 and whose sum is 4. The necessary integers are –1 and 5. Remember to write the common factor 3a as part of the answer. 3a a 2 – 15a= 3a(a – 1)(a + 5)

Slide Factoring by Bottoms Up Multiple a(c) and replace in c Factor (x + 4) (x – 3) Divide each factor by the coefficient a (x + ) (x - ) Reduce if possible (x + ) (x - ) Do not leave as fractions, bring the “bottoms up” (3x + 2) (2x – 1)

Slide EXAMPLE 5 Factor each polynomial. a. 10x x + 3 b. 6r r – 5 = (5x + 1)(2x + 3)a. 10x x + 3 b. 6r r – 5= (2r + 5)(3r – 1)

Slide EXAMPLE 6 Factor –2p 2 – 5p First factor out –1, then proceed. = –1(2p 2 + 5p – 12)–2p 2 – 5p + 12 = –1(p + 4)(2p – 3) = – (p + 4)(2p – 3)

Slide EXAMPLE 7 Factor 4m 3 + 2m 2 – 6m. First, factor out the GCF, 2m. = 2m(2m 2 + m – 3) Look for two integers whose product is 2(–3) = –6 and whose sum is 1. The integers are 3 and –2. = 2m(2m 2 + 3m – 2m – 3) = 2m[m(2m + 3) – 1(2m + 3)] = 2m[(2m + 3)(m – 1)] = 2m(2m + 3)(m – 1)

Slide EXAMPLE 8 Factor 8(z + 5) 2 – 2(z + 5) – 3. Now replace x with z + 5. = 8x 2 –2x – 3 = (2x + 1)(4x – 3) Let x = z (z + 5) 2 – 2(z + 5) – 3 = [2(z + 5) + 1][4(z + 5) – 3] = (2z )(4z + 20 – 3) = (2z + 11)(4z + 17)

Slide EXAMPLE 9 Factor 6r 4 – 13r Let x = r 2. = 6(r 2 ) 2 – 13r = 6x 2 – 13x + 5 = (3x – 5)(2x – 1)Factor. = (3r 2 – 5)(2r 2 – 1) x = r 2