8 7 FACTORING SPECIAL CASES Factoring A process

8. 7: FACTORING SPECIAL CASES: Factoring: A process used to break down any polynomial into simpler polynomials.

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

GOAL:

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

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

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

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

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

SOLUTION: since there is no b term, then b = 0 and we still look at c: az 2+bz+c b= 0 z 2 -16 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: 2 16 x -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: 2 ax +c 2 16 x -81 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: (4 x-9)(4 x+9)

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

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 we still follow the factoring procedure: 2 ax +c 24 x 2 -6 6(4 x 2 -1) 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(2 x-1)(2 x+1)

REAL-WORLD: The area of a square rug is given by 2 4 x -100. 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: 2 ax +c 4 x 2 -100 4(x 2 -25) 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)

VIDEOS: Factoring Quadratics Factoring with GCF: http: //www. khanacademy. org/math/algebra/quadratics/ factoring_quadratics/v/factoring-trinomials-with-acommon-factor Factoring perfect squares: http: //www. khanacademy. org/math/algebra/quadratics/ factoring_quadratics/v/factoring-perfect-squaretrinomials

VIDEOS: Factoring Quadratics Factoring with GCF: http: //www. khanacademy. org/math/algebra/quadratics/f actoring_quadratics/v/u 09 -l 2 -t 1 -we 1 -factoring-specialproducts-1

CLASSWORK: Page 514 -516: Problems: 1, 2, 3, 9, 13, 16, 22, 27, 30, 32, 37, 45.