6.2 Factor Trinomials

Factoring trinomials means rewriting a three-term quadratic expression as a product of two binomials. This skill comes up constantly when solving quadratic equations and simplifying algebraic fractions, so it's worth getting comfortable with the different methods.

Three main approaches cover most situations: the product-sum method for simple trinomials, the ac method for trinomials where the leading coefficient isn't 1, and substitution for expressions with more complex terms.

Factoring Trinomials

Product-Sum Method for Trinomials

This method works for trinomials in the form $x^2 + bx + c$, where the leading coefficient is 1.

1. Find two numbers that multiply to $c$ and add to $b$
2. Use those numbers to split the middle term: rewrite as $x^2 + mx + nx + c$, where $m + n = b$ and $mn = c$
3. Factor by grouping
4. Factor out the common binomial

Example: Factor $x^2 + 7x + 12$

1. Find two numbers that multiply to 12 and add to 7. That's 3 and 4.
2. Rewrite: $x^2 + 3x + 4x + 12$
3. Group and factor: $x(x + 3) + 4(x + 3)$
4. Factor out $(x + 3)$: $(x + 3)(x + 4)$

You can always check by FOILing the result: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. It matches, so you're good.

AC Method for Complex Trinomials

When the leading coefficient $a \neq 1$, the product-sum method doesn't directly apply. The ac method extends the same logic to trinomials in the form $ax^2 + bx + c$.

1. Multiply $a \times c$ to get the product $ac$
2. Find two numbers that multiply to $ac$ and add to $b$
3. Use those two numbers to split the middle term
4. Factor by grouping

Example: Factor $2x^2 + 7x + 3$

1. $ac = 2 \times 3 = 6$
2. Find two numbers that multiply to 6 and add to 7. That's 6 and 1.
3. Rewrite: $2x^2 + 6x + 1x + 3$
4. Group and factor: $2x(x + 3) + 1(x + 3)$
5. Factor out $(x + 3)$: $(2x + 1)(x + 3)$

Notice in step 4 that you factor $2x$ out of the first group and $1$ out of the second group. Both groups must produce the same binomial factor, $(x + 3)$, for the method to work. If they don't match, try a different grouping of your split terms.

Check: $(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$. ✓

Substitution in Trinomial Factoring

Sometimes a trinomial has a repeated expression that makes it look complicated. Substitution lets you temporarily replace that expression with a single variable, factor the simpler version, then swap back.

1. Identify a common term or expression that appears in a pattern
2. Substitute a new variable (like $u$) for that expression
3. Factor the simplified trinomial using the product-sum or ac method
4. Replace $u$ with the original expression

Example: Factor $9x^2 - 12x + 4$

Notice that $9x^2 = (3x)^2$ and $12x = 4(3x)$, so the expression follows a pattern in terms of $3x$.

1. Let $u = 3x$. The trinomial becomes $u^2 - 4u + 4$.

2. Factor: find two numbers that multiply to 4 and add to $-4$. That's $-2$ and $-2$.

3. So $u^2 - 4u + 4 = (u - 2)(u - 2) = (u - 2)^2$

4. Replace $u$: $(3x - 2)^2$

This is also a perfect square trinomial, which is a pattern worth recognizing on its own.

Choosing the Right Factoring Technique

Before you start any factoring method, always check for a Greatest Common Factor (GCF) first. Factor it out, then look at what remains.

After pulling out any GCF, identify the form of the trinomial:

1. $x^2 + bx + c$ form (leading coefficient is 1): use the product-sum method
2. $ax^2 + bx + c$ form where $a \neq 1$: use the ac method
3. Complex or repeated expressions: consider substitution to simplify before factoring

Whatever method you use, always multiply your factors back out to verify the result equals the original trinomial. This takes 30 seconds and catches most mistakes.