Factoring rewrites a polynomial as a product. It is multiplication run backwards: (x + 3)(x + 4) multiplies out to x² + 7x + 12, so factoring x² + 7x + 12 gives (x + 3)(x + 4). You need it to solve quadratics, simplify fractions and find where a graph crosses the x-axis.
The trick is knowing which method to try. Use this order every time.
The order to follow
- Take out the greatest common factor (GCF). Always first.
- Count the terms of what’s left.
- Two terms: difference of squares, or sum or difference of cubes.
- Three terms: a trinomial. Check for a perfect square, then use the methods below.
- Four terms: factor by grouping.
- Check each factor to see whether it factors again.
- Multiply back to check.
Step 1: The greatest common factor
Find the largest number and the highest power of each variable that divides every term.
Example Factor 6x³ + 9x²
- The largest number dividing 6 and 9 is 3. The highest power of x in both terms is x².
- The GCF is 3x². Divide each term by it: 6x³/3x² = 2x and 9x²/3x² = 3.
- Answer: 3x²(2x + 3).
Skipping this step makes everything after it harder. 2x² + 14x + 24 looks awkward, but 2(x² + 7x + 12) is easy.
Trinomials with a leading 1: x² + bx + c
Find two numbers that multiply to c and add to b.
Example Factor x² + 7x + 12
Pairs that multiply to 12: 1 and 12, 2 and 6, 3 and 4. Only 3 and 4 add to 7. Answer: (x + 3)(x + 4).
Example Factor x² − 2x − 15
We need a product of −15 and a sum of −2. The numbers are −5 and 3. Answer: (x − 5)(x + 3).
The signs tell you a lot before you start:
| c | b | The two numbers |
|---|---|---|
| Positive | Positive | Both positive |
| Positive | Negative | Both negative |
| Negative | Either | One positive, one negative |
Trinomials with a leading number: the ac method
For ax² + bx + c with a ≠ 1, guessing gets slow. The ac method turns it into grouping.
Example Factor 6x² + 11x − 10
- Multiply a and c: 6 × (−10) = −60.
- Find two numbers that multiply to −60 and add to 11: 15 and −4.
- Split the middle term with them: 6x² + 15x − 4x − 10.
- Group in pairs and factor each: 3x(2x + 5) − 2(2x + 5).
- Factor out the shared binomial: (3x − 2)(2x + 5).
- Check: (3x − 2)(2x + 5) = 6x² + 15x − 4x − 10 = 6x² + 11x − 10. ✓
Perfect square trinomials
If the first and last terms are perfect squares and the middle term is twice their product, the trinomial is a square:
- a² + 2ab + b² = (a + b)²
- a² − 2ab + b² = (a − b)²
Example Factor 4x² − 12x + 9
4x² = (2x)² and 9 = 3². Twice their product is 2 · 2x · 3 = 12x, which matches the middle term. Answer: (2x − 3)².
Difference of squares
a² − b² = (a − b)(a + b). Look for two perfect squares separated by a minus sign.
Example Factor 49x² − 25
49x² = (7x)² and 25 = 5². Answer: (7x − 5)(7x + 5).
Example Factor x⁴ − 16
- x⁴ = (x²)² and 16 = 4², so this is (x² − 4)(x² + 4).
- x² − 4 is another difference of squares: (x − 2)(x + 2).
- Answer: (x² + 4)(x − 2)(x + 2).
A sum of squares like x² + 4 does not factor using real numbers. Leave it.
Sum and difference of cubes
- a³ − b³ = (a − b)(a² + ab + b²)
- a³ + b³ = (a + b)(a² − ab + b²)
A memory aid for the signs is SOAP: Same sign as the original, Opposite sign, Always Positive.
Example Factor x³ − 8
x³ − 2³ = (x − 2)(x² + 2x + 4).
Example Factor 27x³ + 1
(3x)³ + 1³ = (3x + 1)(9x² − 3x + 1).
Factoring by grouping
With four terms, group them in pairs, factor each pair, and look for a shared factor.
Example Factor x³ + 3x² − 4x − 12
- Group: (x³ + 3x²) + (−4x − 12).
- Factor each group: x²(x + 3) − 4(x + 3).
- Factor out (x + 3): (x + 3)(x² − 4).
- x² − 4 is a difference of squares: (x + 3)(x − 2)(x + 2).
Factor completely
“Factor completely” means keep going until nothing factors further.
Example Factor 2x³ − 18x completely
- GCF: 2x(x² − 9).
- Difference of squares: 2x(x − 3)(x + 3).
Check by multiplying
Multiplying your factors back out takes less time than factoring did, and it proves the answer. If you get the original polynomial, you’re done.
Try a few
Factor x² − 11x + 24.
Show answer
(x − 3)(x − 8). The numbers multiply to 24 and add to −11.
Factor 3x² − 27 completely.
Show answer
3(x − 3)(x + 3). Take out 3, then factor the difference of squares.
Factor 2x² + 7x + 3.
Show answer
(2x + 1)(x + 3). ac = 6, and 6 and 1 add to 7: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3).
Factor 10x² − 13x − 3.
Show answer
(5x + 1)(2x − 3). ac = −30, and −15 and 2 add to −13: 10x² − 15x + 2x − 3 = 5x(2x − 3) + 1(2x − 3).
Factor x³ − 5x² + 2x − 10.
Show answer
(x − 5)(x² + 2). Grouping gives x²(x − 5) + 2(x − 5).
Factor 64x³ − 125.
Show answer
(4x − 5)(16x² + 20x + 25). This is (4x)³ − 5³.
Factor 5x⁴ − 80 completely.
Show answer
5(x² + 4)(x − 2)(x + 2). Take out 5, then x⁴ − 16 factors twice.
Practice
Questions students ask
What if nothing works?
Some polynomials don’t factor over the integers. x² + x + 1 is one: no two integers multiply to 1 and add to 1. For a quadratic, check with the discriminant b² − 4ac. If it isn’t a perfect square, the quadratic won’t factor nicely, and you should use the quadratic formula instead.
Does the order of the factors matter?
No. (x + 3)(x − 2) and (x − 2)(x + 3) are the same.
How does this help solve equations?
If a product equals zero, one of the factors must be zero. Factoring x² + 7x + 12 = 0 into (x + 3)(x + 4) = 0 gives x = −3 or x = −4. See How to Solve Quadratic Equations for the full method.