Factor quadratic expressions, including GCF, difference of squares, and trinomials, to reveal zeros and equivalent forms (Ohio A-SSE.3a, A-APR).

What this topic is asking

Ohio standard A-SSE.3a asks you to factor a quadratic to reveal its zeros. Factoring is the reverse of multiplying, and it is the gateway to solving quadratics by the zero-product property. The patterns, GCF, difference of squares, and trinomial factoring, stack: you check for them in order. None of these patterns is on the reference sheet, so they must be memorized. This is a core skill in the Expressions and Equations reporting category and appears on Part 1.

Always start with the GCF

Before any other pattern, factor out the greatest common factor. This often turns a messy expression into a clean special pattern.

For $3x^2 - 12$: the GCF is $3$, giving $3(x^2 - 4)$, and $x^2 - 4$ is a difference of squares, so $3(x - 2)(x + 2)$.

Difference of squares

If the expression is one perfect square minus another, factor with the fixed pattern.

For $x^2 - 25 = (x - 5)(x + 5)$. Recognize perfect squares: $25 = 5^2$, $49 = 7^2$, $100 = 10^2$, and $4x^2 = (2x)^2$.

Factoring a monic trinomial

A monic trinomial has leading coefficient $1$: $x^2 + bx + c$. Find two numbers that multiply to $c$ and add to $b$.

The signs follow a rule: if $c > 0$, both numbers share the sign of $b$; if $c < 0$, the numbers have opposite signs.

Factoring a non-monic trinomial

When $a \neq 1$, use the $ac$-method: multiply $a \cdot c$, find two numbers that multiply to $ac$ and add to $b$, split the middle term, and factor by grouping.

How Ohio examines this topic

• Equation response. Type the complete factorization.
• Multiple choice. Choose the complete factorization, with "stopped too early" and "dropped the GCF" distractors.
• Drag and drop. Build the factored form from given binomials.

Every factorization can be checked by expanding, which is the reliable way to catch a sign error on a no-calculator part.

Why "completely" is the word that matters

The test almost always asks you to factor completely, and partial factorizations are a frequent trap. "Completely" means continuing until no remaining factor can be broken down further. The classic miss is stopping at $2(x^2 - 9)$ when $x^2 - 9$ is still a difference of squares, or factoring a trinomial but leaving a shared numerical factor inside a binomial. A safe routine is: pull the GCF, apply a special pattern or trinomial method, then re-examine each new factor to see whether it factors again. Only when every factor is irreducible are you done. This discipline is exactly what separates a fully credited answer from a "close but not complete" one on an exact-match item.

How factoring connects to the graph

Factoring is worth the effort because the factors hand you the zeros of the parabola $y = ax^2 + bx + c$ for free. If $x^2 - 7x + 12 = (x - 3)(x - 4)$, then the parabola crosses the $x$-axis at $x = 3$ and $x = 4$, since the output is zero exactly when one factor is zero. So a single factorization answers "what are the solutions?", "where are the $x$-intercepts?", and "what are the zeros of the function?" all at once. That is why factoring threads from this expressions module into solving quadratics and into graphing.

Try this

Q1. Factor $x^2 + 9x + 20$ completely. [1 point]

• Cue. $4 \cdot 5 = 20$, $4 + 5 = 9$, so $(x + 4)(x + 5)$.

Q2. Factor $3x^2 - 27$ completely. [2 points]

• Cue. GCF $3$: $3(x^2 - 9) = 3(x - 3)(x + 3)$.