Table of Contents
In factoring, special products are algebraic expressions that follow very recognizable patterns. Knowing these patterns allows you to factor (and expand) quickly without doing all intermediate multiplication.
This chapter focuses on three main families:
- Square of a binomial
- Difference of squares
- Product of a sum and difference
You should already be familiar with basic polynomial operations and simple factoring of common factors from the parent chapters.
Square of a binomial
A binomial is an expression with two terms, like $x+3$ or $2a-5$.
When you square a binomial, there is a consistent pattern:
- Square of a sum:
$$(a + b)^2 = a^2 + 2ab + b^2$$ - Square of a difference:
$$(a - b)^2 = a^2 - 2ab + b^2$$
Notice:
- The first term is always the square of the first part: $a^2$.
- The last term is always the square of the second part: $b^2$.
- The middle term is twice the product of the two parts: $\pm 2ab$.
- Plus if the binomial has a plus: $(a+b)^2$
- Minus if the binomial has a minus: $(a-b)^2$
Recognizing $(a+b)^2$ and $(a-b)^2$ when factoring
To factor a quadratic expression as a square of a binomial, look for:
- First and last terms are perfect squares.
- The middle term matches $\pm 2$ times the product of the square roots of those squares.
Example:
- $x^2 + 6x + 9$
- $x^2$ is $(x)^2$
- $9$ is $(3)^2$
- Middle term: $6x = 2 \cdot x \cdot 3$
This matches the pattern $a^2 + 2ab + b^2$ with $a=x$, $b=3$, so:
$$x^2 + 6x + 9 = (x + 3)^2$$
- $4y^2 - 12y + 9$
- $4y^2$ is $(2y)^2$
- $9$ is $(3)^2$
- Middle term: $-12y = -2 \cdot 2y \cdot 3$
This matches $a^2 - 2ab + b^2$ with $a=2y$, $b=3$, so:
$$4y^2 - 12y + 9 = (2y - 3)^2$$
Be careful: not every expression with square first and last terms is a perfect square trinomial. The middle term must fit the $2ab$ pattern.
Example:
- $x^2 + 4x + 4 = (x + 2)^2$ (works: middle is $2\cdot x\cdot 2 = 4x$)
- $x^2 + 5x + 4$ is not a square of a binomial (middle is $5x$, not $2\cdot 1\cdot 2 = 4x$)
Difference of squares
Another very common special product involves the difference of two squares.
Pattern for expanding:
$$(a - b)(a + b) = a^2 - b^2$$
In factoring, we use this in reverse:
$$a^2 - b^2 = (a - b)(a + b)$$
Key features:
- Exactly two terms.
- A minus sign between them.
- Each term is a perfect square.
Example:
- $x^2 - 9$
- $x^2 = (x)^2$
- $9 = (3)^2$
So:
$$x^2 - 9 = (x - 3)(x + 3)$$
- $25y^2 - 16$
- $25y^2 = (5y)^2$
- $16 = (4)^2$
So:
$$25y^2 - 16 = (5y - 4)(5y + 4)$$
Recognizing difference of squares
Look for:
- Two terms only.
- A subtraction: $a^2 - b^2$.
- Both terms are perfect squares; that is, you can write each as something squared.
Not a difference of squares:
- $x^2 + 9$ (sum of squares, not difference)
- $x^2 - 8$ ($8$ is not a perfect square)
- $3x^2 - 12$ (first factor out the common factor if possible, then check again)
For example, with $3x^2 - 12$:
First factor out the greatest common factor (GCF):
$$3x^2 - 12 = 3(x^2 - 4)$$
Now $x^2 - 4$ is a difference of squares:
$$x^2 - 4 = (x - 2)(x + 2)$$
So the fully factored form is:
$$3x^2 - 12 = 3(x - 2)(x + 2)$$
Product of a sum and difference
Closely related to difference of squares is the product of a sum and difference:
$$(a + b)(a - b) = a^2 - b^2$$
This is actually the same pattern as before, just written in product form.
When factoring:
- If you start with $a^2 - b^2$, you factor it as $(a + b)(a - b)$.
- If you already have $(a + b)(a - b)$ (or $(a - b)(a + b)$), you could expand it to $a^2 - b^2$.
Example:
- $(2x + 5)(2x - 5)$
Here $a = 2x$, $b = 5$:
$$(2x + 5)(2x - 5) = (2x)^2 - 5^2 = 4x^2 - 25$$
- Factoring $4x^2 - 25$:
- $4x^2 = (2x)^2$
- $25 = (5)^2$
So:
$$4x^2 - 25 = (2x + 5)(2x - 5)$$
Note: $(a + b)(a - b)$ and $(a - b)(a + b)$ are the same up to order; multiplication is commutative.
Recognizing and using special products when factoring
When you factor a polynomial, it is often helpful to check for special product patterns after simpler steps like taking out a common factor:
- Look for a common factor and factor it out.
- Check what remains:
- Is it a difference of squares?
- Is it a perfect square trinomial (square of a binomial)?
Example:
Factor $x^4 - 10x^2 + 25$.
- Treat $x^2$ as a single variable (say $u = x^2$).
- Then $x^4 - 10x^2 + 25$ becomes $u^2 - 10u + 25$.
- $u^2 - 10u + 25$ is a perfect square trinomial:
$$u^2 - 10u + 25 = (u - 5)^2$$ - Substitute back $u = x^2$:
$$(x^2 - 5)^2$$
So:
$$x^4 - 10x^2 + 25 = (x^2 - 5)^2$$
You could then check if $x^2 - 5$ can be factored further over the numbers you are using (over real numbers, $x^2 - 5$ is a difference of squares only if you allow $\sqrt{5}$).
Another example:
Factor $2x^3 - 18x$.
Step 1: Factor out the GCF:
$$2x^3 - 18x = 2x(x^2 - 9)$$
Step 2: Recognize difference of squares in $x^2 - 9$:
$$x^2 - 9 = (x - 3)(x + 3)$$
So:
$$2x^3 - 18x = 2x(x - 3)(x + 3)$$
Here, recognizing $x^2 - 9$ as a difference of squares is the special product step.
Summary of patterns
For quick reference:
- Square of a sum:
$$(a + b)^2 = a^2 + 2ab + b^2$$ - Square of a difference:
$$(a - b)^2 = a^2 - 2ab + b^2$$ - Difference of squares:
$$a^2 - b^2 = (a - b)(a + b)$$
Being able to spot these patterns makes factoring (and expanding) polynomials faster and more reliable, and they appear frequently in later algebra topics.