How to Factor Polynomials: Every Method with Examples

Factoring is the inverse of multiplying. Where multiplying takes \((x + 2)(x + 3)\) and gives you \(x^2 + 5x + 6\), factoring goes the other direction: you take \(x^2 + 5x + 6\) and recover the \((x + 2)(x + 3)\).

It is the single most-tested skill in Algebra I and Algebra II — and it shows up on the SAT, ACT, GRE, ASVAB, and every college placement test. This guide walks through every method you need, in the order you should try them.

The Factoring Decision Tree

When you see a polynomial, try methods in this order:

1. GCF (Greatest Common Factor) — always first.
2. Number of terms? – 2 terms: try difference of squares, sum/difference of cubes. – 3 terms (trinomial): try \(x^2 + bx + c\) or \(ax^2 + bx + c\) methods. – 4 terms: try grouping.
3. Check if any factor can be factored further.

Train your eye to walk this tree every time.

Method 1: Greatest Common Factor (GCF)

Find the largest number and largest variable power that divides every term, and pull it out.

Example 1

\(6x^2 + 9x = 3x(2x + 3)\)

The GCF is $3x$. Both terms divide by $3x$.

Example 2

\(12x^3 – 18x^2 + 6x = 6x(2x^2 – 3x + 1)\)

The GCF is $6x$. The remaining trinomial may factor further (see Method 3).

Always pull the GCF first. It makes every other method easier.

Method 2: Difference of Squares

Pattern: \(a^2 – b^2 = (a + b)(a – b)\).

Example 3

\(x^2 – 16 = (x + 4)(x – 4)\)

Example 4

\(9x^2 – 25 = (3x + 5)(3x – 5)\)

Example 5 (with GCF first)

\(8x^2 – 50 = 2(4x^2 – 25) = 2(2x + 5)(2x – 5)\)

Note: \(a^2 + b^2\) (a sum of squares) does not factor over the real numbers.

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Method 3: Trinomials with Leading Coefficient 1

Pattern: \(x^2 + bx + c = (x + p)(x + q)\) where \(p + q = b\) and \(pq = c\).

Example 6

Factor \(x^2 + 7x + 12\). Find two numbers that multiply to 12 and add to 7: 3 and 4. \(x^2 + 7x + 12 = (x + 3)(x + 4)\)

Example 7

Factor \(x^2 – 5x + 6\). Two numbers that multiply to 6 and add to $-5$: $-2$ and $-3$. \(x^2 – 5x + 6 = (x – 2)(x – 3)\)

Example 8

Factor \(x^2 + x – 12\). Two numbers that multiply to $-12$ and add to 1: 4 and $-3$. \(x^2 + x – 12 = (x + 4)(x – 3)\)

Sign rules: – If \(c\) is positive, both factors have the same sign as \(b\). – If \(c\) is negative, the factors have opposite signs.

Method 4: Trinomials with Leading Coefficient ≠ 1

Pattern: \(ax^2 + bx + c\) where \(a \neq 1\). Use the AC method:

1. Multiply \(a \cdot c\).
2. Find two numbers that multiply to \(ac\) and add to \(b\).
3. Rewrite the middle term using those two numbers.
4. Factor by grouping.

Example 9

Factor \(2x^2 + 7x + 3\). \(a \cdot c = 2 \cdot 3 = 6\). Two numbers that multiply to 6 and add to 7: 6 and 1. Rewrite: \(2x^2 + 6x + x + 3\). Group: \(2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)\).

Example 10

Factor \(6x^2 – x – 2\). \(a \cdot c = 6 \cdot (-2) = -12\). Two numbers that multiply to $-12$ and add to $-1$: $-4$ and 3. Rewrite: \(6x^2 – 4x + 3x – 2\). Group: \(2x(3x – 2) + 1(3x – 2) = (3x – 2)(2x + 1)\).

Method 5: Factor by Grouping (4 Terms)

Pattern: Group terms in pairs, factor a GCF from each pair, then factor out the common binomial.

Example 11

Factor \(x^3 + 3x^2 + 2x + 6\). Group: \((x^3 + 3x^2) + (2x + 6)\). Factor each: \(x^2(x + 3) + 2(x + 3)\). Common binomial: \((x + 3)(x^2 + 2)\).

Example 12

Factor \(2x^3 – 4x^2 + 3x – 6\). Group: \((2x^3 – 4x^2) + (3x – 6)\). Factor: \(2x^2(x – 2) + 3(x – 2)\). Result: \((x – 2)(2x^2 + 3)\).

Method 6: Sum and Difference of Cubes

Patterns: – \(a^3 + b^3 = (a + b)(a^2 – ab + b^2)\) – \(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\)

Example 13

Factor \(x^3 – 8\). \(a = x\), \(b = 2\). Difference of cubes: \((x – 2)(x^2 + 2x + 4)\).

Example 14

Factor \(27x^3 + 1\). \(a = 3x\), \(b = 1\). Sum of cubes: \((3x + 1)(9x^2 – 3x + 1)\).

Method 7: Perfect Square Trinomials (Shortcut)

Patterns: – \(a^2 + 2ab + b^2 = (a + b)^2\) – \(a^2 – 2ab + b^2 = (a – b)^2\)

Example 15

\(x^2 + 6x + 9 = (x + 3)^2\) — because the middle is \(2 \cdot x \cdot 3\).

Example 16

\(4x^2 – 12x + 9 = (2x – 3)^2\).

Putting It All Together

Always start with GCF. Then count the terms. Then try the matching pattern. Then check if any factor can be factored further.

Big Example

Factor completely: \(3x^4 – 48\).

Step 1: GCF. Pull out 3: \(3(x^4 – 16)\). Step 2: Two terms, difference of squares. \(x^4 – 16 = (x^2 + 4)(x^2 – 4)\). Step 3: Continue. \(x^2 – 4\) is also a difference of squares: $(x + 2)(x – 2)$. \(x^2 + 4\) does not factor over the reals. Final answer: \(3(x^2 + 4)(x + 2)(x – 2)\).

Common Mistakes

Skipping the GCF

\(6x^2 + 9x\) is not “two terms — try difference of squares.” Pull the GCF first: $3x(2x + 3)$.

Sign errors in trinomials

\(x^2 – x – 12\) — students often forget that \(-4 + 3 = -1\) and write $(x + 4)(x – 3)$ as $(x – 4)(x + 3)$.

Forgetting to factor completely

\(x^4 – 81 = (x^2 + 9)(x^2 – 9)\) — but \(x^2 – 9\) factors further to $(x + 3)(x – 3)$.

Confusing sum of squares with difference of squares

\(x^2 + 16\) does not factor over the reals. Only the difference factors.

Sign mistakes in sum/difference of cubes

The middle term of the trinomial factor is the opposite sign of the binomial. \(x^3 – 8 = (x – 2)(x^2 + 2x + 4)\).

How to Practice

1. Drill GCF every day for one week — 10 problems daily.
2. Master \(x^2 + bx + c\) trinomials — these show up everywhere.
3. Add the AC method when you’re comfortable.
4. Mix all methods — grab a random worksheet and identify the pattern before solving.
5. Always factor completely. Don’t stop early.

Free Resources

Effortless Math has a complete factoring library:

• Algebra 1 Worksheets — factoring problems by method.
• Math Topics Library — every factoring topic explained.
• Algebra 1 eBooks — full Algebra workbooks.

Frequently Asked Questions

Why do we factor? To solve equations (set each factor equal to zero) and to simplify expressions. Factoring is the gateway to quadratics, rational expressions, and most of Algebra II.

What if nothing factors? Some polynomials are prime — they cannot be factored over the integers. If your AC method fails to find two valid numbers, the polynomial is likely prime.

Should I always check my factoring? Yes — multiply your factors back out. If you get the original, you’re correct.

Is there a shortcut for factoring trinomials? Practice. After 50 trinomials, your brain pattern-matches the factor pairs instantly.

Does factoring work for higher degrees? Yes — methods like grouping and substitution extend factoring to degree 4 and above. Synthetic division and the rational root theorem are next steps in Algebra II.

What’s the most common factoring on the SAT? Difference of squares and simple trinomials. Drill those first.

Factor Like a Pro

Factoring rewards pattern recognition. Walk the decision tree every time. Drill until each method takes under 60 seconds. Then watch your scores rise on every algebra-heavy test you take.

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