Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

Let $Q(x)=x^{2}+2x+3$ , and suppose that $P(x)$ is a polynomial such that $P(Q(x))=x^{6}+6x^{5}+18x^{4}+32x^{3}+35x^{2}+22x+8$ . Compute $P(2)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that $Q(-1)=2$ . Therefore, $P(2)=P(Q(-1))=1-6+18-32+35-22+8=2$ .

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