Factoring means writing an expression as a product of factors; the four core methods are GCF, grouping, trinomial factoring, and difference of squares, and you always check for a GCF first.
Factoring is the reverse of expanding: instead of multiplying factors to get a polynomial, you split a polynomial back into the factors that produce it. The main factoring methods are the greatest common factor, grouping, factoring trinomials, and the difference of two squares, and each fits a particular shape of expression.
What Is Factoring?
Factoring (or factorisation) is the process of writing a polynomial as a product of simpler polynomials, called its factors. It is the exact reverse of expanding: expanding turns $(x + 2)(x + 3)$ into $x^2 + 5x + 6$, and factoring turns $x^2 + 5x + 6$ back into $(x + 2)(x + 3)$.
A factored expression is useful because a product exposes information a sum hides — above all, the values that make the expression zero. That is why factoring is the first move in solving most quadratic and polynomial equations.
The Main Factoring Methods
Factoring is not one procedure but a small toolkit, and each method matches a particular shape of expression. The table below is the map: check for a GCF first, then pick the row that fits whatever remains.
Method | Pattern it fits | Formula or result |
|---|---|---|
Greatest common factor (GCF) | Every term shares a factor | $ab + ac = a(b + c)$ |
Difference of squares | Two terms, both perfect squares, minus sign | $a^2 - b^2 = (a + b)(a - b)$ |
Sum of cubes | Two terms, both perfect cubes, plus sign | $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ |
Difference of cubes | Two terms, both perfect cubes, minus sign | $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ |
Trinomial (leading coefficient 1) | $x^2 + bx + c$ | two numbers multiplying to $c$, adding to $b$ |
Trinomial (AC method) | $ax^2 + bx + c$ with $a \neq 1$ | split the middle term, then group |
Grouping | Four terms | pair them, factor each pair, extract the shared binomial |
A perfect-square trinomial is a special trinomial case: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$.
Why Factoring Methods Exist — "Turning a Sum Into a Product you can Solve"
Factoring is not busywork; it is what makes an equation solvable. A product equals zero only when one of its factors is zero — the zero-product property — so turning $x^2 + 5x + 6 = 0$ into $(x + 2)(x + 3) = 0$ immediately hands you the solutions $x = -2$ and $x = -3$. Without factoring, that equation stays locked.
The methods matter in three concrete places:
Solving quadratic and polynomial equations, where factoring reveals the roots directly.
Simplifying rational expressions, where a shared factor in numerator and denominator cancels.
Modeling with polynomials in physics and engineering, where the factored form shows where a quantity hits zero (a projectile landing, a break-even point).
The historical thread runs back through the algebraists who first treated equations as objects to be manipulated rather than solved case by case. The same identities — the difference of squares and the trinomial patterns — sit inside the broader family of algebraic identities and connect to factorization of algebraic expressions more generally. For quadratics specifically, factoring links directly to factorization of quadratic polynomials.
When to Use Each Method
Work in this order every time:
GCF first, always. Pull out any common factor before looking at patterns. It is the cheapest check and often turns a messy expression into a recognizable one.
Then count the terms that remain.
Two terms — test for a difference of squares, or a sum or difference of cubes.
Three terms — factor the trinomial (numbers multiplying to $c$ and adding to $b$ when the leading coefficient is 1; the AC method when it is not).
Four terms — group into pairs.
Factor completely. After the first pass, check whether any factor can still be broken down — a leftover difference of squares or a further common factor.
Examples of Factoring Methods
Example 1
Factor $6x^3 + 9x^2$ using the greatest common factor.
Find the GCF of the terms. Both share 3 and $x^2$.
$\text{GCF} = 3x^2$
Divide each term by the GCF and write the product.
$6x^3 + 9x^2 = 3x^2(2x + 3)$
Check by expanding: $3x^2 \cdot 2x + 3x^2 \cdot 3 = 6x^3 + 9x^2$.
Example 2 (Wrong path first)
Factor $x^2 + 5x + 6$.
A first instinct is to split it down the middle as $(x + 5)(x + 6)$, matching the numbers you see.
Expand to check.
$(x + 5)(x + 6) = x^2 + 11x + 30$
That is not the original expression, so the guess is wrong. The 5 and 6 were the wrong numbers to place.
The correct method for a trinomial $x^2 + bx + c$ is to find two numbers that multiply to $c$ and add to $b$. Here they multiply to 6 and add to 5, which are 2 and 3.
$x^2 + 5x + 6 = (x + 2)(x + 3)$
Expanding confirms it: $x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.
Example 3
Factor $x^2 - 49$ using the difference of squares.
Recognize both terms as perfect squares with a minus between them.
$x^2 - 49 = x^2 - 7^2$
Apply $a^2 - b^2 = (a + b)(a - b)$ with $a = x$ and $b = 7$.
$x^2 - 49 = (x + 7)(x - 7)$
The pattern only works for a difference; $x^2 + 49$ does not factor over the real numbers.
Example 4
Factor $x^3 + 2x^2 + 5x + 10$ by grouping.
Group the four terms into two pairs.
$(x^3 + 2x^2) + (5x + 10)$
Factor the GCF from each pair.
$x^2(x + 2) + 5(x + 2)$
The shared factor $(x + 2)$ comes out.
$(x + 2)(x^2 + 5)$
Grouping is the go-to method whenever you have four terms.
Example 5
Factor the harder trinomial $2x^2 + 7x + 3$ using the AC (grouping) method.
Multiply the leading coefficient by the constant: $a \times c = 2 \times 3 = 6$.
Find two numbers that multiply to 6 and add to 7: those are 6 and 1.
Split the middle term.
$2x^2 + 6x + x + 3$
Group and factor each pair.
$2x(x + 3) + 1(x + 3)$
Pull out the common $(x + 3)$.
$(x + 3)(2x + 1)$
When the leading coefficient is not 1, splitting the middle term and grouping is the reliable route.
Example 6
Factor $12x^2 - 27$ completely by combining methods.
First pull out the GCF, which is 3.
$12x^2 - 27 = 3(4x^2 - 9)$
The bracket is now a difference of squares: $4x^2 = (2x)^2$ and $9 = 3^2$.
$3(4x^2 - 9) = 3(2x + 3)(2x - 3)$
Factoring is rarely one method — GCF first, then whatever pattern the remaining expression shows.
Common Mistakes With Factoring Methods
Mistake 1: Skipping the GCF and starting with a pattern
Where it slips in: Jumping straight to difference of squares on $12x^2 - 27$ and missing the shared 3.
Don't do this: Writing $12x^2 - 27$ as a difference of squares directly — 12 and 27 are not perfect squares, so the pattern seems not to fit.
The correct way: Always pull the GCF first. After removing 3 you get $4x^2 - 9$, which is a clean difference of squares.
Mistake 2: Wrong signs when the numbers multiply to a negative
Where it slips in: Factoring $x^2 - x - 6$ and reaching for two positive numbers.
Don't do this: Using 2 and 3 as $(x + 2)(x + 3)$; that gives $+6$ and $+5x$, not $-6$ and $-x$.
The correct way: You need numbers that multiply to $-6$ and add to $-1$: that is $-3$ and $+2$, giving $(x - 3)(x + 2)$. The memorizer who recalls "2 and 3" without checking signs lands here; the fix is to test the sum and the sign every time.
Mistake 3: Trying to factor a sum of squares over the reals
Where it slips in: Treating $x^2 + 16$ like $x^2 - 16$.
Don't do this: Writing $x^2 + 16 = (x + 4)(x - 4)$; expanding gives $x^2 - 16$, the wrong sign.
The correct way: The difference-of-squares pattern needs a minus sign. A sum of squares like $x^2 + 16$ does not factor over the real numbers — recognize when to stop.
Conclusion
The four core factoring methods are the GCF, grouping, factoring trinomials, and the difference of squares.
Always check for a GCF first, then choose the method by the number of terms: 2 terms → difference of squares, 3 terms → trinomial, 4 terms → grouping.
Factoring turns a sum into a product, which is what lets you solve equations through the zero-product property.
To practice these methods with guided feedback, work with an algebra tutor, take structured algebra classes, or explore general math classes online.
Practice Questions
Factor each expression completely, then check your answers.
$8x^3 + 12x^2$
$x^2 + 9x + 20$
$x^2 - 81$
$x^3 + 3x^2 + 4x + 12$
$3x^2 + 10x + 8$
$x^3 - 27$
Answers:
GCF $4x^2$: $4x^2(2x + 3)$.
Two numbers multiplying to 20 and adding to 9 are 4 and 5: $(x + 4)(x + 5)$.
Difference of squares: $(x + 9)(x - 9)$.
Grouping: $x^2(x + 3) + 4(x + 3) = (x + 3)(x^2 + 4)$.
AC method ($3 \times 8 = 24$; use 6 and 4): $3x^2 + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)$.
Difference of cubes: $(x - 3)(x^2 + 3x + 9)$.
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