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

What is the easiest way to factor polynomials? Follow the same order every time. First pull out the greatest common factor. Then count the terms: two terms suggest a difference of squares or cubes, three terms suggest a trinomial, and four terms suggest grouping. For example, 3x² − 27 becomes 3(x² − 9), and then 3(x − 3)(x + 3). Finish by checking that no factor can be broken down further.

How do you factor polynomials with 4 terms? Use factoring by grouping. Split the polynomial into two pairs, factor the GCF out of each pair, then factor out the shared binomial. For x³ + 3x² + 2x + 6, group as (x³ + 3x²) + (2x + 6), which gives x²(x + 3) + 2(x + 3). The common factor is (x + 3), so the answer is (x + 3)(x² + 2).

How do you factor polynomials when a is not 1? Use the ac method. Multiply a and c, find two numbers with that product that add to b, split the middle term, and factor by grouping. For 2x² + 7x + 3, ac = 6, and 6 and 1 add to 7. Rewrite as 2x² + 6x + x + 3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Multiply back to check.

What is GCF in factoring polynomials? GCF stands for greatest common factor: the largest number and highest power of each variable that divides every term. In 6x³ + 9x², every term is divisible by 3 and by x², so the GCF is 3x² and the polynomial factors as 3x²(2x + 3). Pulling out the GCF first makes every other factoring method easier, so it should always be your first step.

Is factoring polynomials Algebra 1 or Algebra 2? Both. Algebra 1 usually covers the GCF, difference of squares and trinomials like x² + 5x + 6, often in the unit on quadratics. Algebra 2 builds on those skills with sums and differences of cubes, higher degree polynomials, grouping with harder expressions, and the factor theorem. A solid grasp of the Algebra 1 methods makes the Algebra 2 work much smoother.

How do you factor polynomials completely? Keep factoring until every factor is prime, meaning none can be factored further. After pulling out a GCF, look again at what is left. For 2x² − 18, the GCF gives 2(x² − 9), and x² − 9 is a difference of squares, so the complete answer is 2(x − 3)(x + 3). Stopping at 2(x² − 9) would be only partly factored.

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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