HMMT November, 2012

Algebra Difficulty 4.5 AIME Find the answer United States

Problem:

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

Solution:

Note that $Q(-1) = 2$ . Therefore, $P(2) = P(Q(-1)) = (-1)^{6} + 6(-1)^{5} + 18(-1)^{4} + 32(-1)^{3} + 35(-1)^{2} + 22(-1) + 8 = 1 - 6 + 18 - 32 + 35 - 22 + 8 = 2$ .

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