SAT Mc Graw Hill 2011

CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 317

Factoring

To factor means to write as a product (that is,

a multiplication). All of the terms in a product

are called factors (divisors) of the product.

Example:
There are many ways to factor 12: 12 ×1, 6 ×2,
3 ×4, or 2 × 2 ×3.

Therefore, 1, 2, 3, 4, 6, and 12 are the factorsof 12.

Know how to factor a number into prime factors,

and how to use those factors to find greatest com-

mon factors and least common multiples.

Example:
Two bells, Aand B,ring simultaneously, then bell
Arings every 168 seconds and bell Brings every
360 seconds. What is the minimum number of
secondsbetween simultaneous rings?

This question is asking for the least common multiple
of 168 and 360. The prime factorization of 168 is 2 × 2
× 2 × 3 ×7 and the prime factorization of 360 is 2 × 2 ×
2 × 3 × 3 ×5. A common multiple must have all of the
factors that each of these numbers has, and the small-
est of these is 2 × 2 × 2 × 3 × 3 × 5 × 7 =2,520. So they
ring together every 2,520 seconds.

When factoring polynomials, think of “distribu-

tion in reverse.” This means that you can check

your factoring by distributing, or FOILing, the

factors to make sure that the result is the original

expression. For instance, to factor 3x^2 − 18 x, just

think: what common factor must be “distributed”

to what other factor to get this expression? An-

swer: 3x(x−6) (Check by distributing.) To factor

z^2 + 5 z−6, just think: what two binomials must be

multiplied (by FOILing) to get this expression?

Answer: (z−1)(z+6) (Check by FOILing.)

The Law of FOIL:

=(a)(c+d) +(b)(c+d) (distribution)
=ac+ad+bc+bd (distribution)
First +Outside +Inside +Last
Example:
Factor 3x^2 − 18 x.

Common factor is 3x: 3x^2 − 18 x= 3 x(x−6) (check by
distributing)

Factor z^2 + 5 z−6.

z^2 + 5 z− 6 =(z−1)(z+6) (check by FOILing)

Factoring Formulas

To factor polynomials, it often helps to know

some common factoring formulas:

Difference of squares: x^2 −b^2 =(x+b)(x−b)

Perfect square

trinomials: x^2 + 2 xb+b^2 =(x+b)(x+b)

x^2 − 2 xb+b^2 =(x−b)(x−b)

Simple

trinomials: x^2 +(a+b)x+ab=(x+a)(x+b)

Example:

Factor x^2 −36.

This is a “difference of squares”:

x^2 − 36 =(x−6)(x+6).

Factor x^2 − 5 x−14.

This is a simple trinomial. Look for two numbers that

have a sum of −5 and a product of −14. With a little

guessing and checking, you’ll see that −7 and 2 work.

So x^2 − 5 x− 14 =(x−7)(x+2).

The Zero Product Property

Factoring is a great tool for solving equations

if it’s used with the zero product property,

which says that if the product of a set of num-

bers is 0, then at least one of the numbers in

the set must be 0.

Example:

Solve x^2 − 5 x− 14 =0.

Factor: (x−7)(x+2) = 0

Since their product is 0, either x− 7 =0 or x+ 2 =0, so

x=7 or −2.

The only product property is the zero product

property.

Example:

(x−1)(x+2) = 1 does notimply that x− 1 =1. This

would mean that x=2, which clearly doesn’t work!

Lesson 5: Factoring

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(a+b)(c+d)

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