8.7: FACTORING SPECIAL CASES: Factoring: A process used to break down any polynomial into simpler polynomials.

FACTORING ax 2 + bx + c Procedure: 1) Always look for the GCF of all the terms 2) Factor the remaining terms – pay close attention to the value of coefficient a and follow the proper steps. 3) Re-write the original polynomial as a product of the polynomials that cannot be factored any further.

FACTORING : Case 1: (a+b) 2 ↔ (a+b)(a+b)↔ a 2 +ab+ab+b 2 Case 2: (a-b) 2 ↔ (a-b)(a-b)↔ a 2 –ab-ab+b 2 Case 3: (a+b)(a-b) ↔ a 2 +ab-ab -b 2 ↔ a 2 - b 2 ↔ a 2 +2ab+b 2 ↔ a 2 -2ab+b 2

GOAL:

FACTORING: A perfect square trinomial Ex: What is the FACTORED form of: x 2 -18x+81?

SOLUTION: since the coefficient is 1, we follow the process same process: x 2 -18x+81 ax 2 +bx+c  b= -18  c = +81 Look at the factors of c:  c = +81 : (1)(81), (-1)(-81) (9)(9), (-9)(-9) Take the pair that equals to b when adding the two integers. We take (-9)(-9) since = -18= b Factored form : (x-9)(x-9) = (x-9) 2

YOU TRY IT: Ex: What is the FACTORED form of: x 2 +6x+9?

SOLUTION: since the coefficient is 1, we follow the process same process: X 2 +6x+9 ax 2 +bx+c  b= +6  c = +9 Look at the factors of c:  c = +9 : (1)(9), (-1)(-9) (3)(3), (-3)(-3) Take the pair that equals to b when adding the two integers. We take (3)(3) since 3+3 = +6 = b Factored form : (x+3)(x+3) = (x+3) 2

FACTORING: A Difference of Two Squares Ex: What is the FACTORED form of: z 2 -16?

SOLUTION: since there is no b term, then b = 0 and we still look at c: z az 2 +bz+c  b= 0  c = -16 Look at the factors of c:  c = -16 : (-1)(16), (1)(-16) (-2)(8), (2)(-8), (-4)(4) Take the pair that equals to b when adding the two integers. We take (-4)(4) since 3-4 = 0 = b Thus Factored form is : (z-4)(z+4)

YOU TRY IT: Ex: What is the FACTORED form of: 16x 2 -81?

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 in the x 2, we must look at the a and c coefficients: 16x ax 2 +c  a= +16  c =-81 Look at the factors of a and c:  a : (4)(4)c: (-9)(9) We now see that the factored form is: (4x-9)(4x+9)

YOU TRY IT: Ex: What is the FACTORED form of: 24x 2 -6?

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 we still follow the factoring procedure: 24x 2 -6 ax 2 +c  a= +4  c =-1 Look at the factors of a and c:  a : (2)(2)c: (-1)(1) We now see that the factored form is: 6(2x-1)(2x+1)  6(4x 2 -1)

REAL-WORLD: The area of a square rug is given by 4x What are the possible dimensions of the rug?

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 we still follow the factoring procedure: 4x ax 2 +c  a= +1  c =-25 Look at the factors of a and c:  a : (1)(1)c: (-5)(5) We now see that the factored form is: 4(x-5)(x+5)  4(x 2 -25)

VIDEOS: Factoring Quadratics Factoring perfect squares: actoring_quadratics/v/factoring-perfect-square- trinomials Factoring with GCF: actoring_quadratics/v/factoring-trinomials-with-a- common-factor

VIDEOS: Factoring Quadratics ctoring_quadratics/v/u09-l2-t1-we1-factoring-special- products-1 Factoring with GCF:

CLASSWORK: Page : Problems: 1, 2, 3, 9, 13, 16, 22, 27, 30, 32, 37, 45.