3.4B Factoring Polynomials
Objectives:
A.APR.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A.APR.4: Prove polynomial identities and use them to describe numerical relationships.
A.APR.3: Identify zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial.
For the Board: You will be able to use the Factor Theorem to determine factors of a polynomial.
You will be able to factor the sum and difference of two cubes.
Instruction:
Some polynomials can be factored using FOIL.
Example: Factor x4 – 81.
(x2 + 9)(x2 – 9)
(x2 + 9)(x – 3)(x + 3)
White Board Activity:
Practice: Factor x4 – 13x2 – 36.
(x2 – 9)(x2 – 4)
(x – 3)(x + 3)(x – 2)(x + 2)
Some polynomials can be factored using grouping.
Open the book to page 175 and read example 2:
Example: Factor x3 – x2 – 25x + 25
(x3 – x2) – (25x – 25)
x2(x – 1) – 25(x – 1)
(x2 – 25)(x – 1)
(x – 5)(x + 5)(x – 1)
White Board Activity:
Practice: Factor each expression.
a. x3 – 2x2 – 9x + 18
(x3 – 2x2) – (9x – 18)
x2(x – 2) – 9(x – 2)
(x – 2)(x2 – 9)
(x – 2)(x – 3)(x + 3)
b. 2x3 + x2 + 8x + 4
(2x3 + x2) + (8x + 4)
x2(2x + 1) + 4(2x + 1)
(2x + 1)(x2 + 4)
There are two special cubes that can be factored:
Sum of Two cubes
a3 + b3 = (a + b)(a2 – ab + b2)
Difference of Two cubes
a3 – b3 = (a – b)(a2 + ab + b2)
Open the book to page 175 and read example 3.
Example: Factor each expression.
a. x3 – 64
(x – 4)(x2 + 4x + 16)
b. 125d3 – 8
(5d)3 – 23 = (5d – 2)(25d2 + 10d + 4)
c. 4x4 + 108xy3
4x(x3 + 27y3) = 4x(x3 + 33y3) = 4x(x + 3y)(x2 – 3xy + 9y2)
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d. x – 1
(x2)3 – 1 = (x2 – 1)(x4 + x2 + 1) = (x + 1)(x – 1)(x4 + x2 + 1)
White board Activity:
Practice: Factor each expression.
a. 64 + z3
43 + z3 = (4 + z)(16 – 4z + z2)
b. a3 – 27b3
a3 – 33b3 = (a – 3b)(a2 + 3ab + 9b2)
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c. 2x – 16x2
2x2(x3 – 8) = 2x2(x3 – 23) = 2x2(x – 2)(x2 + 2x + 4)
d. 125a6 + 8
53(a2)3 + 23 = (5a2 + 2)(25a4 – 10a2 + 4)
Assessment:
Question student pairs
Independent Practice:
Text: pgs. 177 – 178 prob. 4 – 15, 20 – 31, 33 – 38.
For a Grade:
Text: pgs. 177 – 178 prob. 20, 26, 28, 38.