Equivalent Expressions

Unit 2 · Advanced Math

What the SAT asks here

Two expressions are equivalent if they produce the same value for every input. On the Digital SAT this is one of the highest-frequency Advanced Math skills — about 3–4 problems per test. You'll factor polynomials, simplify rational expressions, and match transformed expressions to their canonical form.

Common factoring patterns

Distribute (reverse: factor out a GCF)

a(b + c) = ab + ac

Difference of squares

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

Perfect-square trinomial

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

Factor

ax^2 + bx + c

x^2 + 5x + 6 = (x + 2)(x + 3)

Cancelling in rational expressions

Cancel only common factors, never terms across + or −

\frac{(x - 2)(x + 3)}{(x + 3)} = x - 2

SAT Tip: Pull a GCF first, then look for difference-of-squares or trinomial factoring. The SAT often hides one extra factor behind a missed GCF.

Caution: You can only cancel factors, not summands. (x + 3) / 3 is NOT equal to x.

Type 1

Factor a polynomial

Recognise common patterns (GCF, difference of squares, perfect-square trinomial, factor ax² + bx + c) and factor completely.

Example 1

Factor

4x^2 - 36

completely.

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Type 2

Simplify rational expressions

Factor the numerator and denominator separately, then cancel any common factors. State the restriction on x for the cancellation to be valid.

Example 2

Simplify

(x^2 - 4) / (x^2 + 5x + 6)

for x ≠ −2 and x ≠ −3.

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Type 3

Identify the equivalent form

Given an expression, manipulate it (factor, expand, or cancel) and pick the matching equivalent form.

Example 3

Which expression is equivalent to

(x^2 + 6x + 9) / (x + 3)

for x ≠ −3?

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