Warm-ups Find each product. 1. (x – 2)(2x + 7) 2. (3y + 4)(2y + 9) 3. (3n – 5)(n – 7) Factor each trinomial. 4. x 2 +4x – z z h 2 – 17h + 72

Factoring ax 2 + bx + c

3x² + 5x - 2 This is a little harder. I would start with the “first”: (3x )(x ) Your “last” needs to multiply to give you a -2 (#’s will have different signs) Place them in a way that your “outside” and “inside” combine to get 5. OR Now multiply “a” and “c” (3 and –2). You get negative six. Your “outside” answer and your “inside” answer multiply to get negative six AND combine (subtract because the signs must be different) to get 5. Try 6 and –1. You must place numbers so that the “inner” and “outer” products will be 6 and –1. (3x - 1)(x + 2)

( X + )( x + ) = ax 2 + bx + c So, to factor a 2 + bx + c, check the factors of a and the factors of c in the binomials. The sum of the products of the outer and inner terms should be b. Sum of outer and inner products = b Product = c Product = a Since you need to check all the factors of a and the factors of c, it may be helpful to make a table. Then check the products of the outer and inner terms to see if the sum is b. You can multiply the binomials to check your answer.

Example 2A: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x x + 21 ( x + )( x + ) a = 2 and c = 21, Outer + Inner = 17. (x + 7)(2x + 3) Factors of 2 Factors of 21 Outer + Inner 1 and 2 1 and 211(21) + 2(1) = 23 1 and 2 21 and 11(1) + 2(21) = 43 1 and 2 3 and 71(7) + 2(3) = 13 1 and 2 7 and 31(3) + 2(7) = 17    Check (x + 7)(2x + 3) = 2x 2 + 3x + 14x + 21 = 2x x + 21 Use the Foil method.

Factor each trinomial. Check your answer. 9x 2 – 15x + 4 a = 9 and c = 4, Outer + Inner = – 15. Factors of 9 Factors of 4 Outer + Inner 3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 3 and 3 – 2 and – 2 3(–2) + 3(–2) = –12 3 and 3 – 4 and – 13(–1) + 3(– 4)= –15  (3x – 4)(3x – 1) Check (3x – 4)(3x – 1) = 9x 2 – 3x – 12x + 4 = 9x 2 – 15x + 4 Use the Foil method. ( x + )( x + )

3x 2 – 19x + 20 Factor

8x x + 9 Factor:

Another Way—The Box When factoring trinomials, you could use the box again. Put the first term in the top left of a 2 by 2 box. Put the last term in the bottom right square.

Example 1 Factor 6x² + 13x – 5

Try these… Factor each trinomial. Check your answer. 1. 5x x x 2 + 5x – x 2 – 23x x x x 2 - 7x + 3