How to Factor Polynomials: A Complete Step-by-Step Guide
Factoring polynomials means rewriting a polynomial as a product of simpler polynomials. Instead of leaving an expression like x² + 5x + 6, you recover (x + 2)(x + 3).
That one skill unlocks a lot of algebra: solving equations, simplifying rational expressions, finding zeros of functions, and graphing. If you remember only one rule from this guide, make it this:
Always factor out the greatest common factor first. Then count the terms and choose the method that matches the structure.
Quick answer: the factoring decision tree
Use this order every time.
- Write the polynomial in standard form (highest degree first).
- Factor out the GCF (include a negative if the leading coefficient is negative).
- Count the remaining terms:
- 2 terms: difference of squares, or sum/difference of cubes.
- 3 terms: perfect-square trinomial, then x² + bx + c, then the AC method for ax² + bx + c.
- 4+ terms: grouping.
- Check whether any factor can be factored again.
- Multiply to confirm.
A sum of squares such as x² + 9 does not factor over the real numbers.
What “factored completely” means
A polynomial is factored completely (over the integers) when you cannot factor it any further using integer coefficients.
Example:x⁴ − 81 = (x² + 9)(x² − 9) = (x² + 9)(x + 3)(x − 3)
Stopping at (x² + 9)(x² − 9) is incomplete, because x² − 9 is still a difference of squares.
If a polynomial cannot be factored further over the integers, it is called prime (or irreducible over the integers).
Example: x² + x + 1.
Method 1: Greatest common factor (always first)
Factoring out a GCF is the reverse of the distributive property:ab + ac = a(b + c)
How to find the GCF
- Numbers: largest integer that divides every coefficient.
- Variables: each variable raised to the smallest exponent that appears in every term.
Example 1: 15x³ − 10x² = 5x²(3x − 2)
Example 2: 12x³ − 18x² + 6x = 6x(2x² − 3x + 1)
The leftover trinomial still factors:2x² − 3x + 1 = (2x − 1)(x − 1)
So the complete factorization is:12x³ − 18x² + 6x = 6x(2x − 1)(x − 1)
Example 3 (negative leading coefficient):
Factor out −1 (or a negative GCF) so the first term inside is positive:−2x³ + 8x² − 6x = −2x(x² − 4x + 3) = −2x(x − 1)(x − 3)
Method 2: Difference of squares
Pattern:a² − b² = (a − b)(a + b)
Both terms must be perfect squares, and the sign must be minus.
Examples:
x² − 16 = (x − 4)(x + 4)
9x² − 25 = (3x − 5)(3x + 5)
8x² − 50 = 2(4x² − 25) = 2(2x − 5)(2x + 5)
x⁴ − 81 = (x² − 9)(x² + 9) = (x − 3)(x + 3)(x² + 9)
Important: a² + b² does not factor over the reals. So x² + 16 stays as it is.
Method 3: Perfect-square trinomials
Patterns:a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Check three things:
- First term is a square.
- Last term is a square.
- Middle term equals 2ab or −2ab.
Examplesx² + 6x + 9 = (x + 3)²4x² − 20x + 25 = (2x − 5)²
If the middle term is not exactly 2ab, it is not a perfect square. Then use the next method.
Method 4: Trinomials of the form x² + bx + c
Find two numbers that:
- multiply to c
- add to b
Then write (x + p)(x + q).Examplex² + 7x + 12Numbers that multiply to 12 and add to 7: 3 and 4.x² + 7x + 12 = (x + 3)(x + 4)Sign shortcuts
- c > 0 and b > 0: both numbers positive.
- c > 0 and b < 0: both numbers negative.
- c < 0: one positive, one negative. The larger absolute value takes the sign of b.
Examplex² − 5x − 6 = (x − 6)(x + 1)because (−6)(1) = −6 and −6 + 1 = −5.
Method 5: The AC method for ax² + bx + c
When the leading coefficient is not 1, use grouping after splitting the middle term.Steps
- Factor out any GCF first.
- Compute a · c.
- Find two numbers that multiply to ac and add to b.
- Split the middle term.
- Factor by grouping.
Example: Factor 2x² + 9x + 10a · c = 20.
Numbers that multiply to 20 and add to 9:
4 and 5.2x² + 4x + 5x + 10 = 2x(x + 2) + 5(x + 2) = (x + 2)(2x + 5)
Example: Factor 6x² + 5x − 4a · c = −24.
Numbers that multiply to −24 and add to 5: 8 and −3.
6x² + 8x − 3x − 4 = 2x(3x + 4) − 1(3x + 4) = (3x + 4)(2x − 1)
Example: Factor 6x² − x − 2a · c = −12.
Numbers: −4 and 3.
6x² − 4x + 3x − 2 = 2x(3x − 2) + 1(3x − 2) = (3x − 2)(2x + 1)
Method 6: Factoring by grouping (4 terms)
Group in pairs, factor each pair, then pull out the common binomial.
Example:3x³ + 6x² + 2x + 4
(3x³ + 6x²) + (2x + 4) = 3x²(x + 2) + 2(x + 2) = (x + 2)(3x² + 2)
If the first grouping fails, try a different pairing, such as first with third, or rearrange terms.
The AC method is grouping in disguise: you create four terms on purpose so grouping works.
Method 7: Sum and difference of cubes
Patterns:a³ + b³ = (a + b)(a² − ab + b²)a³ − b³ = (a − b)(a² + ab + b²)
Use the SOAP sign pattern:
- Same sign as the original binomial
- Opposite sign in the middle of the trinomial
- Always
- Positive last term in the trinomial
Memorize the first few cubes: 1, 8, 27, 64, 125, 216
.Examples:
x³ − 8 = (x − 2)(x² + 2x + 4)27 + 8y³ = (3 + 2y)(9 − 6y + 4y²)
The quadratic factor from a sum or difference of cubes is usually prime.
Do not try to force it into binomials over the reals.
How to factor higher-degree polynomials
After GCF and special patterns, use substitution or the factor theorem.
Substitution example:x⁴ − 5x² + 4
Let u = x²:
u² − 5u + 4 = (u − 1)(u − 4) = (x² − 1)(x² − 4) = (x − 1)(x + 1)(x − 2)(x + 2)
Factor theoremIf P(c) = 0, then x − c is a factor. Possible rational roots are factors of the constant term over factors of the leading coefficient. Test them, then divide.This is the standard path for many cubics that do not fit grouping.
Common mistakes that cost points
- Skipping the GCF. Then everything after that is messier and often incomplete.
- Stopping too early. Always ask, “Can this factor again?”
- Treating a sum of squares like a difference of squares.
- Mixing SOAP signs on cubes.
- Dropping the GCF from the final answer.
- Forgetting that some polynomials are prime.
- Checking by “eyeballing” instead of multiplying the factors back.
Always check by expanding. If you do not get the original polynomial, the factorization is wrong.
Practice problems
Factor completely.
- 8x + 4
- x² − 25
- x² + 8x + 12
- 2x² + 7x + 3
- x³ + 27
- x³ + 3x² + 4x + 12
- 16x⁴ − 1
- 6x² − 7x − 3
Answers
- 4(2x + 1)
- (x − 5)(x + 5)
- (x + 2)(x + 6)
- (2x + 1)(x + 3)
- (x + 3)(x² − 3x + 9)
- (x + 3)(x² + 4)
- (4x² − 1)(4x² + 1) = (2x − 1)(2x + 1)(4x² + 1)
- (3x + 1)(2x − 3)
FAQ: How to factor polynomials
What is the first step in factoring a polynomial?
Factor out the greatest common factor. Do this before grouping, special formulas, or the AC method.
How do you factor a polynomial with 3 terms?
Check for a perfect-square trinomial. If it is not one, use two numbers that multiply to c and add to b when a = 1. If a is not 1, use the AC method.
How do you factor a polynomial with 4 terms?
Try grouping. If that fails, rearrange the terms and group again.
Can every polynomial be factored?
No. Some are prime over the integers, such as x² + 1 or x² + x + 1. Over the complex numbers, more factorizations exist, but school algebra usually stays with integers or reals.
What is the difference between factoring a polynomial and solving an equation?
Factoring rewrites an expression. Solving uses that factorization plus the zero product property: if AB = 0, then A = 0 or B = 0.
Why does factoring matter?
It turns hard products into simple pieces. Those pieces give intercepts, simplified fractions, and faster equation solving.
Final checklist
Before you submit an answer, run this:
- Did I pull out the GCF?
- Did I match the method to the number of terms?
- Did I factor completely?
- Did I keep every factor, including constants?
- Did multiplying give the original polynomial?
Master that checklist and factoring stops feeling like a bag of disconnected tricks. It becomes one repeatable process.