Factoring

What is Factoring?

Factoring is the reverse of multiplying.

When you expand $(x + 2)(x + 3)$, you get $x^2 + 5x + 6$.

Factoring asks: given $x^2 + 5x + 6$, can we get back to $(x + 2)(x + 3)$?

Why Factor?

Factored form makes some things easy:

• Solving equations: if $(x + 2)(x + 3) = 0$
  • Then either $x = -2$ or $x = -3$
• Finding roots: the factors tell you where the polynomial equals zero

Common Factor

The simplest technique. Look for something that divides every term.

$6x^2$ has factors: $2, 3, 6, x, 2x, 3x, 6x, \ldots$

$9x$ has factors: $3, 9, x, 3x, 9x, \ldots$

The greatest common factor is $3x$. Pull it out:

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

Difference of Squares

When you see $a^2 - b^2$, it factors to $(a + b)(a - b)$.

Why does this work?

Multiply it back:

The middle terms cancel out.

Trinomial Factoring

This is the trickiest pattern. For $x^2 + bx + c$, find two numbers that:

• Multiply to $c$
• Add to $b$

Example: $x^2 + 5x + 6$

We need numbers that multiply to $6$ and add to $5$.

Try factors of $6$: $(1, 6)$, $(2, 3)$

• $1 + 6 = 7$ — no
• $2 + 3 = 5$ — yes!

So: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Perfect Square Trinomial

Some trinomials are perfect squares in disguise.

The patterns:

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

How to spot it:

1. First and last terms are perfect squares
2. Middle term is $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$

Example: $x^2 + 6x + 9$

• $x^2$ is a perfect square ($x$)
• $9$ is a perfect square ($3$)
• $6x = 2 \cdot x \cdot 3$ — yes!

So: $x^2 + 6x + 9 = (x + 3)^2$