Factor and simplify polynomial expressions; apply identities (e.g., difference of squares). | Algebra | ACT Mathematics

Look for factoring patterns before you expand or simplify.

Core Idea

Before expanding or doing long algebra, look for structure: a common factor, a trinomial pattern, or an identity such as $𝑎 2 - 𝑏 2 = (𝑎 - 𝑏) (𝑎 + 𝑏)$ .

Understanding

Rule: ACT polynomial questions reward recognition. If you notice a greatest common factor, a perfect-square trinomial, or a difference of squares early, the expression usually becomes short fast.

The most common identity to recognize is difference of squares. Whenever you see two squares being subtracted, factor instead of expanding: $𝑢 2 - 𝑣 2 = (𝑢 - 𝑣) (𝑢 + 𝑣)$ . That move shows up in factoring, solving quadratics, and simplifying later expressions.

Step by Step

Check first for a greatest common factor.
Then ask whether the expression matches a standard pattern such as difference of squares or a factorable trinomial.
Factor completely before deciding whether anything else can simplify.

Misconceptions

Expanding first even when the expression is ready to factor.
Treating a difference of squares like a square trinomial.
Missing a greatest common factor before checking other patterns.

Question

Worked Example

Which expression is equivalent to $49 𝑚 2 - 81$ ?

Select an answer to see the explanation