Factoring is rewriting a polynomial as a product of simpler pieces — the reverse of multiplying out. It's the key skill behind solving equations, simplifying fractions, and finding roots. Here are the four methods that handle almost everything you'll meet.
Method 1: Greatest Common Factor (GCF)
Always do this first. Pull out the largest factor common to every term.
6x³ + 9x² = 3x²(2x + 3)
Here 3x² divides both terms, so it comes out front. Factoring the GCF often makes the leftover expression easy to factor further.
Method 2: Difference of squares
Any expression of the form a² − b² factors into (a + b)(a − b).
x² − 25 = (x + 5)(x − 5)
9x² − 16 = (3x + 4)(3x − 4)
Spotting a perfect square minus a perfect square is one of the fastest wins in algebra.
Method 3: Factoring trinomials (x² + bx + c)
For a trinomial x² + bx + c, find two numbers that multiply to c and add to b.
Factor x² + 7x + 12:
- Multiply to 12, add to 7 → 3 and 4.
- So
x² + 7x + 12 = (x + 3)(x + 4).
Factor x² − x − 6:
- Multiply to −6, add to −1 → −3 and 2.
- So
x² − x − 6 = (x − 3)(x + 2).
If the leading coefficient isn't 1 (like 2x² + 7x + 3), use the AC method or check the result with the quadratic equation calculator, which also gives the roots.
Method 4: Factoring by grouping
When you have four terms, group them in pairs and factor each pair.
x³ + 3x² + 2x + 6
= (x³ + 3x²) + (2x + 6)
= x²(x + 3) + 2(x + 3)
= (x + 3)(x² + 2)
The shared (x + 3) confirms the grouping worked.
A general game plan
- GCF — pull out any common factor first.
- Count the terms:
- Two terms → try difference of squares.
- Three terms → try the trinomial method.
- Four terms → try grouping.
- Factor again if any piece can still be broken down.
- Check by multiplying your factors back out — you should get the original.
Why factoring matters
Once a polynomial is factored, setting each factor to zero gives the solutions. For instance, (x + 3)(x − 4) = 0 means x = −3 or x = 4. That's the bridge from factoring to solving — and you can verify any quadratic's roots instantly with our quadratic equation calculator.
Frequently asked questions
What does it mean to factor a polynomial?
What should I always check for first when factoring?
How do you factor a difference of squares?
How do you factor a trinomial like x² + 7x + 12?
When should I factor by grouping?
How do I check my factoring is correct?
CalcSolver Editorial Team
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