Step 1 - Pull Out the GCF First

Before reaching for anything more advanced, check whether every term in the polynomial shares a common factor. Find the Greatest Common Factor (GCF) and pull it out of each term - it's the fastest step and it usually shrinks the rest of the problem considerably.

6x³ + 9x² = 3x²(2x + 3)

Step 2 - Spot a Difference of Squares

When one perfect square is subtracted from another - the shape a² - b² - it factors immediately into (a + b)(a - b). There's nothing to guess; just recognize the pattern:

x² - 25 = (x + 5)(x - 5)
4x² - 9 = (2x + 3)(2x - 3)

Step 3 - Factor Simple Trinomials (x² + bx + c)

When the leading coefficient equals 1, look for two numbers that multiply to c and add to b. Once you've found that pair, they drop straight into the two binomials:

x² + 5x + 6 → looking for two numbers that multiply to 6 and add to 5
→ 2 and 3 work → (x + 2)(x + 3)

Step 4 - Factor Trinomials with a Leading Coefficient (ax² + bx + c)

Once a isn't 1 anymore, the same idea needs an extra step, known as the AC method: multiply a by c, find two factors of that product that add up to b, and use them to split the middle term so grouping becomes possible:

2x² + 7x + 3
ac = 6 → factor pair: 1 and 6, since 1 + 6 = 7 ✓
= 2x² + x + 6x + 3
= x(2x + 1) + 3(2x + 1)
= (x + 3)(2x + 1)

Step 5 - Sum and Difference of Cubes

Cubes follow their own fixed patterns. They're worth memorizing outright rather than working them out fresh each time:

a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)

Factoring by Grouping

When you're staring at four terms and none of the earlier patterns fit, try splitting the expression into two pairs, factoring each pair on its own, and checking whether a matching binomial appears in both:

x³ + 2x² + 3x + 6
= x²(x + 2) + 3(x + 2)
= (x² + 3)(x + 2)

Double-Check Your Work

After factoring, multiply the pieces back together - FOIL them or distribute term by term - and confirm you land back on the original polynomial. That check takes seconds and catches nearly every mistake. If you want a second opinion fast, SolverCalc's calcsolver tools can expand the factors back out for you instantly.