1954 AMC 12 Problem 32

Attempt Problem 32 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

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32.

The factors of

x^4+64

are:

(x^2+8)^2

(x^2+8)(x^2-8)

(x^2+2x+4)(x^2-8x+16)

(x^2-4x+8)(x^2-4x-8)

(x^2-4x+8)(x^2+4x+8)

Add and subtract

16x^2

to form a difference of squares

Write

x^4+64=(x^2+8)^2-(4x)^2

Using a difference of squares,

\begin{aligned} x^4+64 &=(x^2+8)^2-16x^2\\ &=(x^2-4x+8)\\ &\quad{}\cdot(x^2+4x+8). \end{aligned}

Thus, the correct answer is E.

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