Page 149 - MarceAlgebra Demystified

136 CHAPTER 6 Factoring

p ffiffiffi p ffiffiffi p ffiffiffi p p ffiffiffi p ffiffiffi p ffiffiffi p p ffiffiffi ffiffiffip p ffiffiffi p
ffiffiffi
ffiffiffi
yÞ¼
15: ð x þ yÞð x ffiffiffi xð xÞþ xð yÞþ x y þ yð yÞ
p p ffiffiffi ffiffiffip p ffiffiffi ffiffiffip p
2 ffiffiffi 2
ffiffiffi
¼ð xÞ þ x y x y þð yÞ ¼ x y
Factoring Quadratic Polynomials

We will now work in the opposite direction—factoring. First we will factor
2
quadratic polynomials, expressions of the form ax þ bx þ c (where a is not
2
0). For example x þ 5x þ 6 is factored as ðx þ 2Þðx þ 3Þ. Quadratic poly-
2
nomials whose first factors are x are the easiest to factor. Their factorization
2
always begins as ðx Þðx Þ. This forces the first factor to be x when
the FOIL method is used All you need to do is fill in the two blanks and
decide when to use plus and minus signs. All quadratic polynomials factor
though some do not factor ‘‘nicely.’’ We will only concern ourselves with
‘‘nicely’’ factorable polynomials in this chapter.
If the second sign is minus, then the signs in the factors will be different
(one plus and one minus). If the second sign is plus then both of the signs will
be the same. In this case, if the first sign in the trinomial is a plus sign, both
signs in the factors will be plus; and if the first sign in the trinomial is a minus
sign, both signs in the factors will be minus.

Examples

2
x 4x 5 ¼ðx Þðx þ Þ or ðx þ Þðx Þ

2
x þ x 12 ¼ðx þ Þðx Þ or ðx Þðx þ Þ
2
x 6x þ 8 ¼ðx Þðx Þ
2
x þ 4x þ 3 ¼ðx þ Þðx þ Þ

Practice

Determine whether to begin the factoring as ðx þ Þðx þ Þ,
ðx Þðx Þ,or ðx Þðx þ Þ:
2
1: x 5x 6 ¼