Math Core

So far you've multiplied polynomials to get longer ones. Factoring runs that process backward: it takes a polynomial and rewrites it as a product. Factored form is what you'll use in the next unit to solve quadratic equations, and the first step in almost every factoring problem is pulling out the greatest common factor.

Factoring is un-multiplying

Compare these two statements:

\underbrace{4x(x + 3)}_{\text{factored form}} \;=\; \underbrace{4x^2 + 12x}_{\text{expanded form}}

Reading left to right is multiplying (distributing). Reading right to left is factoring. Both sides are equal; they're just written differently for different jobs.

Definition

Factoring

To factor a polynomial is to write it as a product of simpler polynomials. A polynomial is factored completely when none of its factors (other than monomials) can be factored further.

The greatest common factor of monomials

The greatest common factor (GCF) of two or more monomials is the largest monomial that divides each of them. Find it in two parts:

Numbers: the GCF of the coefficients, as in arithmetic.
Variables: each variable that appears in every term, raised to the smallest exponent that appears.

Why the smallest exponent? $x^2$ divides both $x^2$ and $x^5$ , but $x^3$ doesn't divide $x^2$ . You can't take out more copies of $x$ than the term with the fewest has.

Worked example: Finding a GCF

Find the GCF of $12x^4y^2$ , $18x^3y^5$ and $30x^2y^3$ .

Coefficients: the GCF of $12$ , $18$ and $30$ is $6$ .
$x$ : the exponents are $4$ , $3$ and $2$ . The smallest is $2$ , so $x^2$ .
$y$ : the exponents are $2$ , $5$ and $3$ . The smallest is $2$ , so $y^2$ .

The GCF is $6x^2y^2$ .

Factoring out the GCF

Find the GCF of all the terms.
Divide each term by the GCF. The quotients go inside the parentheses.
Write the answer as $\text{GCF} \times (\text{quotients})$ .
Check by distributing. You should get the original polynomial back.

Worked example: A binomial and a trinomial

Factor $12x^3 + 18x^2$ and $8a^4b^2 - 20a^3b^3 + 4a^2b$ .

For $12x^3 + 18x^2$ : the GCF is $6x^2$ . Divide: $\dfrac{12x^3}{6x^2} = 2x$ and $\dfrac{18x^2}{6x^2} = 3$ . So

12x^3 + 18x^2 = 6x^2(2x + 3).

For $8a^4b^2 - 20a^3b^3 + 4a^2b$ : the GCF of $8$ , $20$ and $4$ is $4$ ; the smallest power of $a$ is $a^2$ ; the smallest power of $b$ is $b^1$ . The GCF is $4a^2b$ .

\frac{8a^4b^2}{4a^2b} = 2a^2b, \qquad \frac{-20a^3b^3}{4a^2b} = -5ab^2, \qquad \frac{4a^2b}{4a^2b} = 1.

So $8a^4b^2 - 20a^3b^3 + 4a^2b = 4a^2b(2a^2b - 5ab^2 + 1)$ .

Common mistake

When a term equals the GCF, its quotient is $1$ , not $0$ . Writing $4a^2b(2a^2b - 5ab^2)$ loses a term; distribute it back and you only get two terms instead of three. Every term of the original must leave something behind in the parentheses.

A negative GCF

When the leading coefficient is negative, it's standard to factor out a negative GCF so the polynomial in parentheses starts with a positive term. Every sign inside flips.

Worked example: Factoring out a negative

Factor $-6x^2 + 15x - 9$ so that the first term in parentheses is positive.

The GCF of the numbers is $3$ ; use $-3$ . Divide each term by $-3$ :

-6x^2 + 15x - 9 = -3(2x^2 - 5x + 3).

Check: $-3 \cdot 2x^2 = -6x^2$ , $-3 \cdot (-5x) = 15x$ , $-3 \cdot 3 = -9$ . ✓

A binomial as the common factor

The common factor doesn't have to be a monomial. In $3x(x - 2) + 5(x - 2)$ , both terms contain the factor $(x - 2)$ . Factor it out just as you would a number:

3x(x - 2) + 5(x - 2) = (x - 2)(3x + 5).

If it helps, temporarily call $(x - 2)$ something simple like $A$ : then $3xA + 5A = A(3x + 5)$ .

Factoring by grouping

A polynomial with four terms often has no GCF for all four, but you can pair the terms and factor each pair. If both pairs leave the same binomial, factor it out.

Worked example: Factoring by grouping

Factor $x^3 + 4x^2 + 3x + 12$ .

\begin{aligned} x^3 + 4x^2 + 3x + 12 &= (x^3 + 4x^2) + (3x + 12) && \text{group in pairs} \\ &= x^2(x + 4) + 3(x + 4) && \text{GCF of each pair} \\ &= (x + 4)(x^2 + 3) && \text{common binomial factor} \end{aligned}

Check: $(x + 4)(x^2 + 3) = x^3 + 3x + 4x^2 + 12$ . ✓

If the second pair starts with a minus sign, factor out a negative so the binomials match. For example, $x^3 - 2x^2 - 5x + 10 = x^2(x - 2) - 5(x - 2) = (x - 2)(x^2 - 5)$ . You'll use grouping again when you factor $ax^2 + bx + c$ .

Tip

Factoring always has a built-in check: multiply your answer back out. It takes a few seconds, and it catches missing terms, wrong signs and GCFs that weren't really the greatest.

Practice

Practice 1

What is the GCF of $18x^4$ and $24x^2$ ?

Practice 2

Factor $10y^2 - 25y$ completely.

Practice 3

When $14x^3 + 21x^2 - 7x$ is factored as $7x(\;\cdots\;)$ , what is the constant term inside the parentheses?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Factor $9a^3b^2 + 12a^2b^3$ completely.

Practice 5

Factor $-4x^2 + 20x - 8$ so that the first term in parentheses is positive.

Practice 6

Factor $2x(x + 5) - 7(x + 5)$ .

Practice 7

Factor $x^3 - 2x^2 + 5x - 10$ by grouping.

Practice 8

A rectangle has area $6x^2 + 9x$ and width $3x$ . Its length is a binomial. What is the length when $x = 4$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.