Factoring trinomials: x2 + bx + c Lesson 72 Factoring trinomials: x2 + bx + c

Factoring trinomials The polynomial x2 + bx + c is a trinomial Like numbers, some trinomials can be factored into 2 linear factors.

Factoring when c is positive Factor x2 + 9x + 18 In this trinomial b is 9 and c is 18 Because b is positive, it must be the sum of 2 positive numbers that are factors of c List all factors of c, which is 18: 1,18 2,9 3,6 Which pair adds up to b , which is 9? 3,6 So x2 +9x + 18 = (x+3)(x+6)

practice Factor x2 + 5x + 6 x2 + 8x + 15 x2 + 4x + 4 x2 + 4x + 3

Factoring when c is positive and b is negative Factor x2-5x+4 List factors of 4: 1,4 2,2 -1, -4 -2,-2 Which pair adds up to -5 ? -1,-4 x2 -5x + 4 = (x-1)(x-4)

practice Factor: x2 – 8x + 15 x2 -7x + 12 x2 -5x + 6 x2 – 4x + 3

Factoring when c is negative Factor x2 + 3x -10 List factors of -10 -1,10 -10,1 -2,5 -5,2 Which pair adds up to +3? -2,5 x2 + 3x -10 = (x-2)(x+5)

practice Factor: x2 – 7x – 8 x2 – 7x – 30 x2 + 3x -54 x2 + 5x - 14

Factoring with 2 variables x2 + 5xy + 6y2 b = 5y c = 6y2 list factors of 6y2 1,6y2 6,y2 2,3y2 3,2y2 y,6y 2y,3y Which pair adds up to 5y 2y, 3y So x2 + 5xy + 6y2 = (x+2y)(x+3y)

factor x2 + 2xy – 3y2 x2 + 4xy +4y2 x2 -7xy -18y2

Rearranging terms before factoring -21-4x+x2 Write the trinomial in standard form , then factor x2 -4x-21 list factors of -21, that add up to -4 -1,21 1,-21 -3,7 3,-7 3, -7 adds up to -4 so x2-4x-21 = (x+3)(x-7)

practice Factor: -24 + 5x + x2 3x – 10 + x2 -8 + x2 – 7x

Evaluating trinomials Evaluate x2+5x-14 and its factors for x = 3 x2 + 5x -14 = (x+7)(x-2) (3)2 +5(3)-14 = (3+7)(3-2) 9 + 15-14 = (10)(1) 10 = 10

practice Evaluate x2 + 5x – 14 for x = 2

Lesson 75 The pattern used for trinomials is different when a is not 1. Factor 2x2 + 7x + 5 2x2 has factors 2x and x so we know (2x )( x ) Factors of 5 are 1,5 -1,-5 Because the middle term is positive, we can eliminate the -1,-5 pair Check each other pair to see if the middle terms added equal the middle term in the trinomial (2x + 1)(x+5) or (2x +5)(x+1) 2x2 +10x+x+5 2x2+2x+5x+5 2x2 +11x +5 2x2 + 7x + 5

factor 3x2 +13x+12 6x2 +11x+3

Factoring when b is negative and c is positive Factor 6x2 - 11x + 3 factors of 6x2 (x )(6x ) or (2x )(3x ) Factors of +3 1,3 or -1,-3 Since the middle term is negative, we can eliminate 1,3 Try (x-1)(6x-3) = 6x2-3x-6x+3=6x2-9x+3 (x-3)(6x-1)=6x2-x-18x+3=6x2-19x+3 (2x-1)(3x-3)=6x2-6x-3x+3=6x2-9x+3 (2x-3)(3x-1)=6x2-2x-9x+3=6x2-11x+3

practice 4x2 -23x + 15 10x2-23x+12

Factoring when c is negative 4x2 + 4x-3 Factors of 4x2 (4x )(x )or(2x )(2x ) Factors of -3 1, -3 or -1,3 (4x+1)(x-3)= 4x2-11x-3 (4x-3)(x+1)= 4x2+x-3 (4x-1)(x+3)= 4x2+11x-3 (4x+3)(x-1)= 4x2-x-3 (2x+1)(2x-3)= 4x2-4x-3 (2x-1)(2x+3)=4x2+4x-3

factor 6x2+8x-8 2x2-3x-20 3x2-11x-4

Factoring with 2 variables Factor 2x2-11xy+5y2 Factors of 2x2 (2x )(x ) Factors of 5y2 5y, y or -5y, -y Since the middle sign is negative , we can eliminate the 5y,y (2x-5y)(x-y) or (2x-y)(x-5y) 2x2-2xy-5xy+5y2 or 2x2-10xy-xy+5y2 2x2-7xy+5y2 or 2x2-11xy+5y2

practice Factor 2x2-5xy+2y2 6x2+11xy+4y2

Rearranging before factoring Trinomials must be in standard form before factoring

Factoring trinomials by using the GCF Lesson 79 Factoring trinomials by using the GCF

Factoring trinomials with positive leading coefficients Always factor the GCF first, then continue factoring as in previous lessons x4+5x3+6x2 GCF is x2, so x2(x2+5x+6) x2(x+3)(x+2)

Practice Factor 4x3-4x2-80x a5+3a4-18a3 2m4-16m3+30m2

Factoring with negative lead coefficients It will always be easier to factor a trinomial with a positive lead coefficient, so whenever possible, factor out a negative if your trinomial has a negative lead coefficient. -x2+x+56= -1(x2-x-56)=-1(x-8)(x+7)