8-7 Factoring Special Cases
Quick Review
When you factor a perfect-square trinomial, the two
binomial factors are the same.
a2 + 2 ab + b2 = (a + b)(a + b) = (a + b)2
a2 - 2 ab + b2 = {a~ b){a - b) = (a - b)2
When you factor a difference of squares of two terms, the
two binomial factors are the sum and the difference of the
two terms.
a2 — b2 = (a + b){a — b)Ex a m p l e
What is the factored form of 8 I t 2 — 90 1 + 25?
First rewrite the first and last terms as squares. Then
determine if the middle term equals - 2ab.
81f2 - 90t + 25 = (91)2 - 90 t + 52
= (9t)2 - 2(90(5) + 52
= (9f- 5)2Ex e r c i s e s
Factor each expression.- s2 - 20s + 100 60. 16q2 + 56q + 49
- r2 — 64 62. 9z2 - 16
- 25m2 + 80m + 64 64. 49a2 - 4
- g2 - 225 66. 9 p2 - 42 p+49
- 36b2 - 12b +1 68. w2 + 24w + 144
- 32n2 - 8 70. 25x2 - 36
- Geometry Find an expression for the length of a
side of a square with an area of 9n2 + 54n + 81. - Reasoning Suppose you are using algebra tiles to
factor a quadratic trinomial. What do you know
about the factors of the trinomial when the tiles form
a square?
8-8 Factoring by Grouping
Quick Review
When a polynomial has four or more terms, you may be
able to group the terms and find a common binomial
factor. Then you can use the Distributive Property to
factor the polynomial.Ex a m p l e
What is the factored form of 2 r 3 - 12r2 + 5r — 30?
First factor out the GCF from each group of two terms. Then
factor out a common binomial factor.
2r3 - 12r2 + 5r - 30 = 2r\r - 6) + 5(r - 6)
= (2r2 + 5)(r-6)V
Ex e r c i se s
Find the GCF of the first two terms and the GCF of the last
two terms for each polynomial.- 6y3 - 3y2 + 2y - 1
- 8m3 + 40m2 + 6m + 1 5
Factor completely. - 6 d 4 + 4d3 — 6 d2 — 4d
- lib3 — 6b2 + lib — 6
- 45z3 + 20z2 + 9z + 4
- 9a3 - 12a2 + 18a - 24
5 3 8 Chapter 8 Ch ap t er Review