Factoring polynomials means breaking down a polynomial expression into simpler expressions (factors) that, when multiplied together, give back the original polynomial. It’s one of the most important skills in algebra because it makes solving equations, simplifying expressions, and graphing functions much easier.
Why Factoring Matters
Before diving into methods, it helps to understand why factoring is useful. Many algebraic problems become simple once a polynomial is factored. For example, solving a quadratic equation like x² + 5x + 6 = 0 is much easier when written as (x + 2)(x + 3) = 0, because you can immediately see that x = -2 or x = -3.
Method 1: Factoring Out the Greatest Common Factor (GCF)
The first step in factoring any polynomial is to check if there’s a common factor in all terms.
Example: 6x² + 12x
Both terms share a common factor of 6x.
6x² + 12x = 6x(x + 2)
Always look for the GCF first, since it simplifies the rest of the factoring process.
Method 2: Factoring Trinomials (ax² + bx + c)
When a = 1, you need two numbers that multiply to c and add up to b.
Example: x² + 7x + 12
You need two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4.
x² + 7x + 12 = (x + 3)(x + 4)
Example with negative numbers: x² – x – 6
You need two numbers that multiply to -6 and add to -1. Those numbers are -3 and 2.
x² – x – 6 = (x – 3)(x + 2)
Method 3: Factoring by Grouping
This method is useful for polynomials with four terms.
Example: x³ + 3x² + 2x + 6
Group the terms: (x³ + 3x²) + (2x + 6)
Factor each group: x²(x + 3) + 2(x + 3)
Since (x + 3) is common: (x + 3)(x² + 2)
Method 4: Difference of Squares
When you see a polynomial in the form a² – b², it factors into (a – b)(a + b).
Example: x² – 16
This is a difference of squares since 16 = 4².
x² – 16 = (x – 4)(x + 4)
Method 5: Perfect Square Trinomials
A perfect square trinomial follows the pattern a² + 2ab + b² = (a + b)² or a² – 2ab + b² = (a – b)².
Example: x² + 6x + 9
Here, a = x, b = 3, and 2ab = 6x, which matches.
x² + 6x + 9 = (x + 3)²
Factoring Trinomials When a ≠ 1
When the leading coefficient isn’t 1, the process takes a bit more work.
Example: 2x² + 7x + 3
Multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7: those are 6 and 1.
Rewrite the middle term: 2x² + 6x + x + 3
Group and factor: 2x(x + 3) + 1(x + 3)
(x + 3)(2x + 1)
Checking Your Work
After factoring, it’s a good habit to multiply the factors back out to confirm they give the original polynomial. Manually checking each step can take time, especially with larger expressions, so many students use Calc Solver to verify their factored answers instantly and catch mistakes before moving on.
Practice Problems
Try factoring these on your own:
- x² + 9x + 20
- x² – 4x – 21
- 4x² – 25
- x² + 10x + 25
- 3x² + 11x + 6
Working through a mix of trinomials, grouping problems, and special patterns like difference of squares will make the process feel much more natural over time. If a problem gets too complex to check by hand, running it through Calc Solver is a quick way to confirm the correct factors before continuing your practice.