Factoring trinomials ax² + bx +c a = 1

x² + bx +c Product is C Sum is B

Factor the trinomials Example: x² + 7x + 12 a = 1 b = 7 c = 12 We need factors of 12 (2 #’s that multiply to 12) that add up to 7 Factors of 12: 12 and 1 2 and 6 3 and 4 Answer: (x+4)(x+3) We must always check our answer by foil, box method, or distributing!!!!! We must ALWAYS SHOW THE CHECK!

Example: x² + 9x +20 a = 1 b = 9 c = 20 Factors of 20: 20 and 1; 2 and 10; 4 and 5 Answer: (x+4)(x+5)

Example: x² -8x +15 A = 1 b = -8 c = 15 Factors of 15: 5 and 3; 15 and 1 None of those add up to -8 Factors of 15: -5 and -3; -15 and -1 Answer: (x-5)(x-3)

Example: x² + 4x – 12 A = 1 b = 4 c = -12 Factors of -12: 12 and -1; -12 and 1; 2 and -6; 6 and -2; 3 and -4; 4 and -3 Answer: (x + 6)(x-2) Example: x² - 3x – 40 A = 1 b = -3 c = -40 Factors of -40: 1 and -40; 40 and -1; 2 and -20; 20 and -2; 4 and -10; 10 and -4; 5 and -8; 8 and -5 Answer: (x – 8)(x +5)

Let’s look for a hint! x² + 9x +20 = (x+4)(x+5) x² -8x +15 = (x-5)(x-3) x² + 4x – 12 = (x + 6)(x-2) x² - 3x – 40 = (x – 8)(x +5) ax² + bx +c = ( + )( + ) ax² - bx +c = ( - ) ( - ) ax² + bx - c = ( + )( - ) ax² - bx – c = ( + )( - )

Practice x² + 9x +18 x² -13x + 22 x² + 5x – 36 x² - x – 42

X² + 4x - 10 When we are unable to find the two factors to add up to the middle term we are unable to factor our polynomial! We call these polynomials prime! A prime polynomial is not factorable!

What happens when there are two variables? Example: x² + 5xy + 6y² Answer: (x + #y)(x + #y) Factors of 6y² that add up to 5y! 1y and 6y or 2y and 3y Answer: (x + 3y)(x + 2y)

Practice a² - 13ab + 30b² x² - 4xy – 77y² (a -3b)(a – 10b) (x -11y)(x + 7y)

When A does not equal 1 Example: 3m² - 24m – 60 Always check for a GCF! GCF: 3 3(m² - 8m – 20) Factor the quotient! 3(m-10)(m + 2)

Practice x³ + 3x² - 4x this is not ax² + bx +c 7x² + 14xy – 21y² 2t⁵ - 14t⁴ + 24t³ X (x² + 3x – 4) X(x +4)(x – 1) 2. 7(x² + 2y – 3y²) 7(x + 3y)(x – 1y) 2t³(t² - 7t + 12) 2t³(t – 4)(t – 3)