Chapter Twelve — Factoring Polynomials
The Humongous Book of Algebra Problems
265
12.19 Factor the expression: 2(x + 1) + 3y(x + 1).
This expression consists of two terms, 2(x + 1) and 3y(x + 1). Both terms have
one common factor, the binomial (x + 1), so the greatest common factor of the
expression is x + 1. Divide both terms by that quantity.
Factor the expression by writing it as the product of the greatest common factor
and the preceding quotients.
2(x + 1) + 3y(x + 1) = (x + 1)(2 + 3y)
Factoring by Grouping
You can factor out binomials, too
12.20 Factor the expression: 9x(2y – 1) + 8y – 4.
This expression consists of three terms: 9x(2y – 1) is the first term, 8y is the
second, and –4 is the third. Notice that the first term is expressed as a product.
The greatest common factor of the second and third terms (8y and –4) is 4.
Factor it out of the expression 8y – 4 to get 4(2y – 1). Rewrite the expression
using the newly factored form of the second and third terms.
The expression contains two terms with a binomial greatest common factor:
2y – 1. Factor the expression using the method described in Problem 12.19.
9x(2y – 1) + 4(2y – 1) = (2y – 1)(9x + 4)
12.21 Factor the expression by grouping: 3x
3
+ 15x
2
+ 2x + 10.
To factor by grouping, group together the polynomial terms that have common
factors. In this problem, 3x
3
and 15x
2
have common factors, and 2x and 10 share
a common factor as well.
(3x
3
+ 15x
2
) + (2x + 10)
The greatest common factor of the left group of terms is 3x
2
; the greatest
common factor of the right group is 2. Write each group in factored form.
3x
2
(x + 5) + 2(x + 5)
If the
expression
was 2w + 3yw,
it’d be easy
to tell that w
was the greatest
common factor;
just plug in (x + 1)
where w is and you
get Problem 12.19.
Common factors
don’t have to be
monomials. They can
be binomials like
this (or trinomials,
or fractions, or
square roots…
the list goes on
and on).
When you
factor (2y – 1)
out of the
expression, you’re
left with 9x in the
rst term and 4 in
the second. Add
those values inside
parentheses (9x + 4)
and multiply by the
greatest common
factor:
(9x + 4)(2y – 1).
The terms
you end up grouping
are usually next to
each other.