Factor polynomial expressions in one variable: a greatest common monomial factor, trinomials of the form ax^2 + bx + c, perfect-square trinomials, and the difference of two squares (A.EO.5).

What this topic is asking

A.EO.5 asks you to factor polynomials in one variable: pull out a greatest common factor, factor a quadratic trinomial, and recognize the two special products (perfect-square trinomial, difference of squares). On the Virginia Algebra I SOL these are Expressions and Operations items, and factoring is also the engine behind solving quadratics later. They appear as multiple choice, fill-in-the-blank (type the factored form), and matching.

Step 1: the greatest common factor

The GCF is the largest factor shared by every term, combining the numerical and variable parts. Pull it out using the distributive property in reverse:

Here the GCF is $3x^2$ ($3$ divides $6$ and $9$; $x^2$ is the lowest power of $x$ present). Factoring out the GCF first often turns a hard-looking polynomial into a simpler one, and on the SOL a factor left inside (failing to take the full GCF) costs the point.

Difference of squares

When a binomial is one perfect square minus another, it factors instantly:

Identify $a$ and $b$ as the square roots of each term: $x^2 - 49 = (x + 7)(x - 7)$, and $9x^2 - 16 = (3x + 4)(3x - 4)$. A sum of squares ($a^2 + b^2$) does not factor over the real numbers.

Perfect-square trinomials

A trinomial that came from squaring a binomial has a recognizable shape: the first and last terms are perfect squares, and the middle term is twice the product of their roots.

So $x^2 + 10x + 25 = (x + 5)^2$ (since $25 = 5^2$ and $10x = 2 \cdot x \cdot 5$).

General trinomials

For $x^2 + bx + c$ with leading coefficient $1$, find two numbers that multiply to $c$ and add to $b$; those numbers go straight into the binomials.

The order to try methods

Factoring is most reliable as a checklist:

1. GCF. Always first.
2. Count the terms. Two terms, check for a difference of squares. Three terms, check for a perfect-square trinomial, then factor the general trinomial.
3. Check by multiplying. FOIL or distribute to confirm you recover the original.

Why factoring is just multiplying in reverse

Every factoring rule is a multiplication rule read backward. $(a + b)(a - b) = a^2 - b^2$ expands a product into a difference of squares; factoring runs the same identity the other way, $a^2 - b^2 = (a + b)(a - b)$. The two-numbers method for trinomials is the inverse of FOIL: when you multiply $(x + p)(x + q)$ you get $x^2 + (p + q)x + pq$, so the middle coefficient is the sum $p + q$ and the constant is the product $pq$. Factoring just asks for $p$ and $q$ given that sum and product. Because the operations are inverses, you can always check a factorization by expanding it, which is why no SOL factoring item should ever be guessed: multiply your answer and see if it matches.

How the SOL examines this topic

• Fill-in-the-blank. Type the complete factorization, for example $(x + 5)(x - 3)$.
• Multiple choice. Pick the correct factored form; distractors come from sign errors and incomplete factoring.
• Matching. Pair polynomials with their factorizations, or identify which trinomial is a perfect square.

Try this

Q1. Factor $x^2 - 9x + 20$. [2 points]

• Cue. Two numbers multiply to $20$, add to $-9$: $-4$ and $-5$, so $(x - 4)(x - 5)$.

Q2. Factor $5x^2 - 20$. [1 point]

• Cue. GCF $5$: $5(x^2 - 4) = 5(x + 2)(x - 2)$.