Which expression is equivalent to x^4 - 81? | SAT Equivalent expressions Question, Answer Explained

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Which expression is equivalent to $x^4 - 81$ ?

$(x - 3)^4$

$(x^2 - 9)^2$

$(x^2 + 9)(x - 3)^2$

$(x^2 + 9)(x + 3)(x - 3)$

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Answer: D

The expression $x^4 - 81$ is a difference of squares: $(x^2)^2 - 9^2 = (x^2 + 9)(x^2 - 9)$ . The factor $x^2 - 9$ is also a difference of squares, $(x + 3)(x - 3)$ . So $x^4 - 81 = (x^2 + 9)(x + 3)(x - 3)$ .

Choice B squares $x^2 - 9$ , but $(x^2 - 9)^2 = x^4 - 18x^2 + 81$ , which has a middle term and a positive constant.

Why the other choices are wrong

A. (x - 3)⁴ expands to x⁴ - 12x³ + 54x² - 108x + 81, which has extra terms and a positive constant.
B. (x² - 9)² = x⁴ - 18x² + 81, which has a middle term and +81 instead of -81.
C. This factors x² - 9 as (x - 3)², but x² - 9 = (x + 3)(x - 3).

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