SAT Equivalent Expressions: Cancel a Factor and Keep the Domain

For x ≠ 4, which expression is equivalent to (x² − 16)/(x − 4)?

Factor the numerator

x² − 16 is a difference of squares: (x − 4)(x + 4).

Cancel the nonzero factor

Because x ≠ 4, the factor x − 4 is nonzero and can be canceled. The result is x + 4.

Retain the original restriction

The original denominator is zero at x = 4. Simplifying does not make the original fraction defined there.

Polynomial division gives quotient x + 4 and remainder 0 because (x − 4)(x + 4) = x² − 16. At x = 6, the original value is (36 − 16)/2 = 10, matching 6 + 4. That substitution checks the result; the factorization establishes the identity.

• x − 4 would multiply back to x² − 8x + 16, not the numerator.
• x would multiply back to x² − 4x.
• x² + 4 would produce a cubic after multiplication by x − 4.

For x ≠ −3, simplify (x² − 9)/(x + 3).

Answer: x − 3, with x ≠ −3

Factor x² − 9 as (x − 3)(x + 3). Cancel x + 3 only where it is nonzero.

If this was difficult, return to equivalent expressions practice and identify the precise step to improve. These are original PeakSAT teaching examples, not official College Board questions or calibrated score predictions. For official adaptive test practice, use Bluebook (opens in a new tab).