8.7: FACTORING SPECIAL CASES:

Factoring: A process used to break down any polynomial into simpler polynomials.

FACTORING ax2 + bx + c Procedure:

1) Always look for the GCF of all the terms2) 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)↔ a2 +ab+ab+b2

Case 2:

(a-b)2 ↔ (a-b)(a-b) ↔ a2 –ab-ab+b2

Case 3:

(a+b)(a-b) ↔ a2 +ab-ab -b2 ↔ a2- b2

↔ a2+2ab+b2

↔ a2-2ab+b2

GOAL:

FACTORING: A perfect square trinomial

Ex: What is the FACTORED form of:

x2-18x+81?

SOLUTION: since the coefficient is 1, we follow the process same process: x2-18x+81 ax2+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 -9+ -9 = -18= bFactored form : (x-9)(x-9) = (x-9)2

YOU TRY IT:

Ex: What is the FACTORED form of:

x2+6x+9?

SOLUTION: since the coefficient is 1, we follow the process same process: X2+6x+9 ax2+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 = bFactored form : (x+3)(x+3) = (x+3)2

FACTORING: A Difference of Two Squares

Ex: What is the FACTORED form of:

z2-16?

SOLUTION: since there is no b term, then b = 0 and we still look at c: z2-16 az2+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 = bThus Factored form is : (z-4)(z+4)

YOU TRY IT:

Ex: What is the FACTORED form of:

16x2-81?

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 in the x2, we must look at the a and c coefficients:

16x2-81 ax2+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:

24x2-6?

SOLUTION: To factor a difference of two squares with a coefficient ≠ 1 we still follow the factoring procedure:

24x2-6 ax2+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(4x2-1)

REAL-WORLD:

The area of a square rug is given by 4x2-100.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:

4x2-100ax2+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(x2-25)

VIDEOS: FactoringQuadratics

Factoring perfect squares:http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoring-perfect-square-trinomials

Factoring with GCF:http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoring-trinomials-with-a-common-factor

VIDEOS: FactoringQuadratics

http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/u09-l2-t1-we1-factoring-special-products-1

Factoring with GCF:

CLASSWORK:

Page 514-516:

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