Math Core

In the special products lesson you learned three patterns for multiplying. Read backward, those same patterns are factoring shortcuts. Once you can recognize a difference of squares or a perfect square trinomial on sight, you can factor it in one step, with no factor pairs to list.

Difference of two squares

Recall that $(a + b)(a - b) = a^2 - b^2$ . Reversed:

a^2 - b^2 = (a + b)(a - b).

To use it, you need a binomial with two perfect squares and a minus sign between them. Perfect squares to recognize include $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$ , as well as even powers of variables like $x^2$ , $x^4$ and $y^6$ (because $x^4 = (x^2)^2$ ).

Worked example: Differences of squares

Factor $x^2 - 49$ and $9a^2 - 25b^2$ .

$x^2 - 49 = x^2 - 7^2$ , so

x^2 - 49 = (x + 7)(x - 7).

$9a^2 = (3a)^2$ and $25b^2 = (5b)^2$ , so

9a^2 - 25b^2 = (3a + 5b)(3a - 5b).

Common mistake

A sum of two squares, like $x^2 + 49$ , does not factor using real numbers. Try it: $(x + 7)(x + 7) = x^2 + 14x + 49$ and $(x + 7)(x - 7) = x^2 - 49$ . Neither gives $x^2 + 49$ . Only the difference of squares follows the pattern.

Perfect square trinomials

The other two special products, read backward, are

a^2 + 2ab + b^2 = (a + b)^2 \qquad\qquad a^2 - 2ab + b^2 = (a - b)^2.

A trinomial is a perfect square trinomial when:

the first and last terms are perfect squares, $a^2$ and $b^2$ , and both are positive;
the middle term is $2ab$ or $-2ab$ .

For $x^2 + 12x + 36$ : $x^2 = (x)^2$ , $36 = 6^2$ , and $2 \cdot x \cdot 6 = 12x$ . ✓ So $x^2 + 12x + 36 = (x + 6)^2$ .

For $x^2 + 13x + 36$ : the ends are still perfect squares, but $13x \ne 12x$ , so it is not a perfect square trinomial. (It happens to factor as $(x + 4)(x + 9)$ by the usual method.)

Special factoring patterns

\begin{aligned} a^2 - b^2 &= (a + b)(a - b) \\ a^2 + 2ab + b^2 &= (a + b)^2 \\ a^2 - 2ab + b^2 &= (a - b)^2 \end{aligned}

The sign of the middle term tells you the sign inside the square.

Worked example: Perfect square trinomials

Factor $4x^2 - 20x + 25$ .

$4x^2 = (2x)^2$ , so $a = 2x$ .
$25 = 5^2$ , so $b = 5$ .
$2ab = 2(2x)(5) = 20x$ , and the middle term is $-20x$ .

So $4x^2 - 20x + 25 = (2x - 5)^2$ .

You could also factor that trinomial with the ac method and get $(2x - 5)(2x - 5)$ . The pattern is simply faster.

Factoring completely

Many polynomials need more than one step. A reliable order:

Factor out the GCF, if there is one.
Count the terms of what's left.
- Two terms: is it a difference of squares?
- Three terms: is it a perfect square trinomial? If not, use product-and-sum or the ac method.
- Four terms: try grouping.
Check each factor to see whether it can be factored again.

Worked example: GCF, then a pattern

Factor $2x^3 - 18x$ completely.

The GCF is $2x$ : $2x^3 - 18x = 2x(x^2 - 9)$ .

$x^2 - 9$ is a difference of squares:

2x^3 - 18x = 2x(x + 3)(x - 3).

Worked example: Factoring twice

Factor $x^4 - 16$ completely.

$x^4 = (x^2)^2$ and $16 = 4^2$ , so first

x^4 - 16 = (x^2 + 4)(x^2 - 4).

The factor $x^2 - 4$ is another difference of squares, while $x^2 + 4$ is a sum of squares and doesn't factor. So

x^4 - 16 = (x^2 + 4)(x + 2)(x - 2).

Tip

The difference of squares also makes some arithmetic easy. $53^2 - 47^2 = (53 + 47)(53 - 47) = 100 \cdot 6 = 600$ , with no squaring required.

Practice

Practice 1

Factor $x^2 - 81$ .

Practice 2

Factor $16y^2 - 1$ .

Practice 3

Which is a perfect square trinomial?

Practice 4

The trinomial $9x^2 - 24x + 16$ can be written as $(ax - b)^2$ with $a$ and $b$ positive. What is $b$ ?

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

Practice 5

Factor $3x^2 - 75$ completely.

Practice 6

For what positive value of $k$ is $x^2 + kx + 64$ a perfect square trinomial?

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

Practice 7

Factor $2x^3 + 12x^2 + 18x$ completely.

Practice 8

Factor $x^4 - 81$ completely.

Practice 9

Use a difference of squares to compute $101^2 - 99^2$ .

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