Factoring Techniques for Polynomials

Factoring rewrites a polynomial as a product of simpler polynomials — the reverse of multiplying them out. Three go-to techniques — GCF, grouping, and special-product patterns like the difference of squares — cover almost every polynomial you'll meet in this course.

By the end you'll be able to look at a polynomial, pick the fastest technique, factor it, and verify your answer by multiplying the factors back out.

Predict: which technique factors \(x^2 - 9\) fastest — GCF, grouping, or a special pattern? Click a technique below, then Verify to check your answer by multiplying the factors back out.

Each button loads its own example polynomial. Pick one to see it factored, then verify the factoring by expanding the factors back to the original.

Pick a technique above to see it applied to an example.

Factoring exposes a polynomial's roots and is the workhorse behind solving equations and sketching graphs — and picking the right technique fast is most of the skill.

Procedural

Work through the techniques in this order, every time:

  • GCF first, always. Factor out the largest monomial common to every term before doing anything else — it shrinks the numbers you're working with for whatever comes next.
  • Four terms → grouping. Pair the terms, factor each pair, then factor out the shared binomial.
  • Two or three terms → check the special-product patterns before grinding through anything else — recognizing a pattern is faster than deriving it.
Formal

Memorize these patterns; recognizing them beats re-deriving them:

  • Difference of squares: \(a^2-b^2=(a-b)(a+b)\)
  • Perfect-square trinomial: \(a^2 \pm 2ab+b^2=(a\pm b)^2\)
  • Sum of cubes: \(a^3+b^3=(a+b)(a^2-ab+b^2)\)
  • Difference of cubes: \(a^3-b^3=(a-b)(a^2+ab+b^2)\)

A sum of squares, \(a^2+b^2\), does not factor over the real numbers — there's no real pair of binomials whose product loses the middle term the way the difference pattern does.

Applied

Fully factored form is what lets you read off a polynomial's zeros directly — each factor \((x-r)\) hands you a root \(x=r\) for free, which is exactly what you need to solve equations or sketch where a graph crosses the x-axis without a calculator.

Worked example

Factor \(p(x) = 3x^4 - 48\) completely.
Step 1 — GCF: \(3x^4 - 48 = 3(x^4-16)\).
Step 2 — difference of squares (\(x^4=(x^2)^2\), \(16=4^2\)): \(3(x^4-16) = 3(x^2-4)(x^2+4)\).
Step 3 — difference of squares again on \(x^2-4\): \(= 3(x-2)(x+2)(x^2+4)\).
\(x^2+4\) is irreducible over the reals, so this is fully factored.

Your turn

Factor \(p(x) = 2x^4 - 32\) completely.
Step 1 — GCF: \(2x^4 - 32 = 2(x^4-16)\).
Step 2 — difference of squares: \(2(x^4-16) = 2(x^2-4)(x^2+4)\).
Now finish it — one more difference of squares is hiding in there. \(x^2-4 =\) ____

Reveal the answer

\(x^2-4 = (x-2)(x+2)\), so \(p(x) = 2(x-2)(x+2)(x^2+4)\) — same structure as the worked example above, just with a GCF of 2 instead of 3. \(x^2+4\) still doesn't factor over the reals, so this is fully factored.

More info — why grouping works for trinomials too

Grouping isn't just for four-term polynomials — it's also how you factor a trinomial \(ax^2+bx+c\) when \(a \neq 1\). Split the middle term \(bx\) into two terms whose coefficients multiply to \(a \cdot c\) and add to \(b\), then group the resulting four terms as usual. For \(2x^2+7x+3\): find two numbers multiplying to \(2\cdot3=6\) and adding to 7 — that's 6 and 1, so \(2x^2+6x+x+3 = 2x(x+3)+1(x+3) = (x+3)(2x+1)\). See the Purplemath link in Dive deeper for more worked splits.

Check your understanding

Question 1 of 4

Factor \(x^2 - 16\) completely.

Question 2 of 4

Why doesn't \(x^2+9\) factor using the difference-of-squares pattern the way \(x^2-9\) does?

Question 3 of 4

To factor \(x^3+5x^2+3x+15\), which technique applies fastest?

Question 4 of 4

You're told \(p(3) = 0\) for \(p(x) = x^2 - x - 6\). Using the Factor Theorem together with factoring, what is the fully factored form of \(p(x)\)?

Recap

  • Factor out the GCF first, always — it simplifies whatever technique comes next.
  • Four terms with no shared GCF → grouping: pair terms, factor each pair, factor out the shared binomial.
  • Two or three terms → check special-product patterns first: difference of squares \(a^2-b^2=(a-b)(a+b)\), perfect-square trinomials, and sum/difference of cubes.
  • A sum of squares, \(a^2+b^2\), never factors over the real numbers.
  • Verify any factoring by multiplying the factors back out — it should reproduce the original polynomial exactly.

Dive deeper

Sources

  • Factoring Techniques for Polynomials