Math Core

Quadratic trinomials like $x^2 + 7x + 12$ are everywhere in algebra: in area problems, in projectile motion, and in the equations of the next unit. When the leading coefficient is $1$ , factoring one comes down to a small number puzzle: find two numbers with a given product and a given sum.

Where the numbers come from

Multiply two binomials of the form $(x + p)(x + q)$ :

(x + p)(x + q) = x^2 + qx + px + pq = x^2 + (p + q)x + pq.

Look at where each coefficient comes from:

the coefficient of $x$ is the sum $p + q$ ;
the constant term is the product $pq$ .

For example, $(x + 3)(x + 4) = x^2 + 7x + 12$ : the $7$ is $3 + 4$ and the $12$ is $3 \cdot 4$ . Factoring reverses this. Given $x^2 + bx + c$ , you look for the two numbers $p$ and $q$ .

Factoring x² + bx + c

To factor $x^2 + bx + c$ , find two integers $p$ and $q$ with

p \cdot q = c \qquad \text{and} \qquad p + q = b.

Then $x^2 + bx + c = (x + p)(x + q)$ .

A good routine: list the factor pairs of $c$ and check the sum of each pair until one equals $b$ .

Worked example: Both signs positive

Factor $x^2 + 7x + 12$ .

You need a product of $12$ and a sum of $7$ .

factor pair of $12$	sum
$1, 12$	$13$
$2, 6$	$8$
$3, 4$	$7$ ✓

So $x^2 + 7x + 12 = (x + 3)(x + 4)$ .

Using the signs

The signs of $b$ and $c$ tell you the signs of $p$ and $q$ before you start listing:

$c$	$b$	signs of $p$ and $q$	example
positive	positive	both positive	$x^2 + 7x + 12 = (x + 3)(x + 4)$
positive	negative	both negative	$x^2 - 9x + 20 = (x - 4)(x - 5)$
negative	either	one positive, one negative	$x^2 + 2x - 15 = (x + 5)(x - 3)$

Here's the reasoning. A positive product means the two numbers have the same sign, and the sum tells you which sign. A negative product means opposite signs, and the number with the larger absolute value takes the sign of $b$ .

Worked example: Positive c, negative b

Factor $x^2 - 9x + 20$ .

Product $20$ (positive) and sum $-9$ (negative), so both numbers are negative. Try negative factor pairs of $20$ : $-1$ and $-20$ sum to $-21$ ; $-2$ and $-10$ sum to $-12$ ; $-4$ and $-5$ sum to $-9$ . ✓

x^2 - 9x + 20 = (x - 4)(x - 5).

Worked example: Negative c

Factor $x^2 + 2x - 15$ and $x^2 - 4x - 21$ .

For $x^2 + 2x - 15$ : the product is $-15$ , so the signs differ. The sum is $+2$ , so the larger number is positive. The pair $5$ and $-3$ works: $5 \cdot (-3) = -15$ and $5 + (-3) = 2$ .

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

For $x^2 - 4x - 21$ : product $-21$ , sum $-4$ , so the larger number is negative. The pair $-7$ and $3$ works.

x^2 - 4x - 21 = (x - 7)(x + 3).

Common mistake

It's easy to find the right two numbers and then attach the wrong signs. $(x - 5)(x + 3)$ looks almost like $(x + 5)(x - 3)$ , but it multiplies out to $x^2 - 2x - 15$ , not $x^2 + 2x - 15$ . Always check the middle term by multiplying the outer and inner products.

GCF first

If all three terms share a common factor, factor it out first. What's left is often an $x^2 + bx + c$ trinomial.

Worked example: Taking out a GCF first

Factor $3x^2 - 6x - 24$ completely.

Every term is divisible by $3$ :

3x^2 - 6x - 24 = 3(x^2 - 2x - 8).

Now factor $x^2 - 2x - 8$ : product $-8$ , sum $-2$ . The pair $-4$ and $2$ works.

3x^2 - 6x - 24 = 3(x - 4)(x + 2).

Don't drop the $3$ . It's part of the complete factorization.

When nothing works

Some trinomials can't be factored using integers. For $x^2 + 5x + 8$ , the factor pairs of $8$ are $1, 8$ (sum $9$ ) and $2, 4$ (sum $6$ ). Neither sums to $5$ , and negative pairs have negative sums. A polynomial that can't be factored this way is called prime. Once you've checked every factor pair, you can say so with confidence.

Tip

When $c$ has many factor pairs, start with the pair whose numbers are closest together if $|b|$ is small, and with the pair that's farthest apart (like $1$ and $c$ ) if $|b|$ is large. Sums grow as the numbers spread apart.

Practice

Practice 1

Factor $x^2 + 8x + 15$ .

Practice 2

Factor $x^2 - 10x + 21$ .

Practice 3

The trinomial $x^2 + x - 30$ factors as $(x + a)(x + b)$ . Find $a$ and $b$ .

Separate answers with commas, e.g. 2, -5

Practice 4

Factor $x^2 - 5x - 24$ .

Practice 5

The trinomial $x^2 - 13x + 36$ factors as $(x - 4)(x - m)$ . What is $m$ ?

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

Practice 6

Which trinomial is prime (cannot be factored using integers)?

Practice 7

Factor $2x^2 + 2x - 40$ completely.

Practice 8

The trinomial $x^2 + bx + 18$ can be factored using integers, and $b$ is positive. What is the largest possible value of $b$ ?

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