The Main Ideas

What Factoring Really Is

Factoring is just distributing in reverse: instead of multiplying pieces together, you break an expression back apart into the pieces that were multiplied. Master a short checklist and you can factor almost anything.

Factoring = un-distributing

\[ ab + ac = a(b + c) \]

A factored expression is written as a product (things multiplied). You are pulling out what the terms share.

Step 1: always pull the GCF first

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

Take out the greatest common factor — the biggest number and variable every term shares — before trying anything else. It makes what is left simpler.

Then count the terms

2 terms → look for a difference of squares
3 terms → factor the trinomial
4 terms → factor by grouping

This checklist tells you which tool to reach for next.

Trinomial, leading coefficient 1

\[ x^2 + bx + c = (x + m)(x + n) \]

Find two numbers that multiply to \(c\) and add to \(b\). Example: \(x^2 + 5x + 6 = (x+2)(x+3)\) because \(2\cdot 3 = 6\) and \(2 + 3 = 5\).

Special pattern: difference of squares

\[ a^2 - b^2 = (a + b)(a - b) \]

Two perfect squares with a minus sign between them. Example: \(x^2 - 16 = (x+4)(x-4)\). (A sum of squares does not factor.)

Special pattern: perfect-square trinomial

\[ a^2 \pm 2ab + b^2 = (a \pm b)^2 \]

The first and last terms are perfect squares and the middle is twice their roots. Example: \(x^2 + 6x + 9 = (x+3)^2\).

Why we factor: it finds the zeros

\[ (x - 2)(x + 5) = 0 \Rightarrow x = 2 \ \text{or}\ x = -5 \]

A product is zero only when a factor is zero (the zero-product property). Factoring is how you solve quadratics and find where a graph crosses the x-axis.

See It Worked Out

Worked Examples

Example 1 · GCF, then difference of squares

Factor completely: \(2x^2 - 18\)

Pull out the GCF, which is 2.
\[ 2x^2 - 18 = 2(x^2 - 9) \]
Now \(x^2 - 9\) is a difference of squares (\(9 = 3^2\)).
\[ x^2 - 9 = (x + 3)(x - 3) \]

\( 2x^2 - 18 = 2(x + 3)(x - 3) \)

Example 2 · A basic trinomial

Factor: \(x^2 - 7x + 12\)

Need two numbers that multiply to \(+12\) and add to \(-7\). Both must be negative.
\(-3\) and \(-4\) work: \((-3)(-4) = 12\) and \(-3 + (-4) = -7\).
Write the factors.
\[ x^2 - 7x + 12 = (x - 3)(x - 4) \]

\( (x - 3)(x - 4) \)

Example 3 · Leading coefficient ≠ 1 (grouping)

Factor: \(6x^2 + 11x + 3\)

Multiply \(a\cdot c = 6 \cdot 3 = 18\). Find two numbers that multiply to 18 and add to 11.
\(9\) and \(2\): \(9 \cdot 2 = 18\), \(9 + 2 = 11\). Split the middle term.
\[ 6x^2 + 9x + 2x + 3 \]
Group in pairs and factor each pair.
\[ 3x(2x + 3) + 1(2x + 3) \]
Both share \((2x + 3)\); pull it out.
\[ (2x + 3)(3x + 1) \]

\( (2x + 3)(3x + 1) \)

Your Turn

Practice Problems

Factor each one, then reveal the solution to check yourself.

\(3x^2 + 12x\)

Show solution

GCF is \(3x\).

\[ 3x(x + 4) \]

\(x^2 + 2x - 15\)

Show solution

Two numbers that multiply to \(-15\) and add to \(2\): \(+5\) and \(-3\).

\[ (x + 5)(x - 3) \]

\(x^2 - 49\)

Show solution

Difference of squares (\(49 = 7^2\)).

\[ (x + 7)(x - 7) \]

\(4x^2 - 12x + 9\)

Show solution

Perfect-square trinomial: \(4x^2 = (2x)^2\), \(9 = 3^2\), and \(2(2x)(3) = 12x\).

\[ (2x - 3)^2 \]

\(2x^2 + 7x + 3\)

Show solution

\(a\cdot c = 6\); \(6\) and \(1\) multiply to 6 and add to 7. Group: \(2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3)\).

\[ (x + 3)(2x + 1) \]

Keep Going

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