Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

$29 \cdot 39 x^{4}+4$ is a factor of
(A) $x^{2}+2$ .
(B) $x+1$ .
(C) $x^{2}-2 x+2$ .
(D) $x^{2}-4$ .
(E) None of the above.
(4th American High School Mathematics Examination, 1953)

This was a multiple-choice question, but the options didn't survive into the source we have. The answer given is C, and the solution below works it through.

Solution

[Solution] $x^{4}+4=x^{4}+4 x^{2}+4-4 x^{2}=\left(x^{2}+2\right)^{2}-4 x^{2}$ $=\left(x^{2}+2 x+2\right)\left(x^{2}-2 x+2\right)$ .

Therefore, the answer is $(C)$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.