Factoring can sometimes feel like you’re being asked to take an algebra expression apart and somehow know exactly where all the pieces belong.
Luckily, certain expressions follow patterns. Once you learn to recognize those patterns, factoring becomes much easier.
For the SAT, three useful patterns to know are the difference of two squares, perfect square trinomials, and the sum and difference of cubes.
The secret isn’t just memorizing the formulas. It’s learning to recognize what kind of expression you’re looking at.
First: What Does “Factoring” Mean?
Factoring is basically undoing multiplication.
For example:
(x + 2)(x + 5)
If we multiply this out, we get:
x² + 7x + 10
Factoring goes in the opposite direction:
x² + 7x + 10 = (x + 2)(x + 5)
On the SAT, factoring can help you simplify expressions, solve equations, identify zeros, and rewrite expressions in more useful forms.
Pattern #1: The Difference of Two Squares
The formula is:
a² − b² = (a − b)(a + b)
This pattern has a few important clues. You need:
- Two terms
- Subtraction between them
- Both terms to be perfect squares
For example:
x² − 25
Since 25 = 5², we can think of this as:
x² − 5²
Now use the difference of two squares pattern:
x² − 25 = (x − 5)(x + 5)
Notice that one factor uses subtraction and the other uses addition.
Quick Memory Trick: Difference of squares becomes “minus and plus.”
Watch for Hidden Perfect Squares
The SAT may not always make the pattern as obvious as x² − 25.
Consider:
9x² − 16
Both terms are still perfect squares because:
9x² = (3x)²
and
16 = 4²
Therefore:
9x² − 16 = (3x − 4)(3x + 4)
The key is recognizing that you’re looking at one square minus another square.
Pattern #2: Perfect Square Trinomials
A trinomial is simply an expression containing three terms.
The perfect square trinomial patterns are:
(a + b)² = a² + 2ab + b²
and
(a − b)² = a² − 2ab + b²
These formulas work in both directions.
For example:
(x + 3)²
can be expanded as:
x² + 6x + 9
So if the SAT gives you:
x² + 6x + 9
you should be able to recognize that it factors as:
(x + 3)²
Why?
The first term is a perfect square:
x² = (x)²
The last term is also a perfect square:
9 = 3²
And the middle term is twice the product of x and 3:
2(x)(3) = 6x
Everything fits the pattern.
The Minus Version Works Too
Consider:
x² − 10x + 25
The first and last terms are perfect squares:
x² = (x)²
and
25 = 5²
The middle term is:
−2(x)(5) = −10x
Therefore:
x² − 10x + 25 = (x − 5)²
SAT Tip: When the first and last terms of a trinomial are perfect squares, check whether the middle term is twice the product of their square roots. You may have found a perfect square trinomial.
Pattern #3: The Sum and Difference of Cubes
There are two formulas to know here.
For the sum of two cubes:
a³ + b³ = (a + b)(a² − ab + b²)
For the difference of two cubes:
a³ − b³ = (a − b)(a² + ab + b²)
These look more complicated, but there’s a helpful pattern hiding in the signs.
For a sum of cubes, the first factor keeps the plus sign:
a³ + b³ → (a + b)(a² − ab + b²)
For a difference of cubes, the first factor keeps the minus sign:
a³ − b³ → (a − b)(a² + ab + b²)
Notice that the middle sign in the second set of parentheses does the opposite.
An Example With Cubes
Suppose you have:
x³ + 8
Since:
8 = 2³
we can think of this as:
x³ + 2³
Using the sum of cubes formula:
x³ + 8 = (x + 2)(x² − 2x + 4)
Now consider:
x³ − 8
Using the difference of cubes formula:
x³ − 8 = (x − 2)(x² + 2x + 4)
The signs change, but the overall structure stays the same.
Don’t Confuse Squares and Cubes
Here’s an important distinction.
The difference of two squares can be factored:
a² − b² = (a − b)(a + b)
But the sum of two squares:
a² + b²
does not factor the same way over the real numbers.
Cubes are different. Both a sum of cubes and a difference of cubes can be factored:
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
That’s an important difference to remember.
Why Does the SAT Care About These Patterns?
The SAT may not simply give you an expression and say, “Factor this.”
Instead, factoring might be one step inside a larger problem.
You might use factoring to:
- Solve an equation
- Find the zeros of a function
- Simplify an algebraic expression
- Determine which expression is equivalent to another
- Understand features of a graph
- Rewrite an equation into a more useful form
That’s why recognizing these patterns quickly can save valuable time.
A Quick Factoring Checklist
When you see an expression that might be factorable, ask yourself:
Do I have two perfect squares with subtraction between them?
Look for:
a² − b² = (a − b)(a + b)
Do I have three terms with perfect squares at the beginning and end?
Check for:
a² + 2ab + b² = (a + b)²
or
a² − 2ab + b² = (a − b)²
Do I have two perfect cubes?
Check for:
a³ + b³
or
a³ − b³
Don’t immediately start doing complicated algebra. First ask yourself whether the expression matches a familiar pattern.
The Three Patterns to Remember
Difference of Two Squares
a² − b² = (a − b)(a + b)
Perfect Square Trinomials
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Sum and Difference of Cubes
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
Final Thoughts
Factoring gets easier when you stop treating every expression like a brand-new puzzle.
Sometimes the SAT is simply dressing up a pattern you’ve already seen.
When you spot two perfect squares being subtracted, think difference of squares. When you see a three-term expression with perfect squares on the ends, check for a perfect square trinomial. And when two perfect cubes are being added or subtracted, think sum or difference of cubes.
Learn to recognize the shapes of these expressions, and factoring becomes less about guessing and more about knowing exactly which tool to use.