Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.2 AIME Find the answer United States

Problem:
Suppose that $p(x)$ is a polynomial and that $p(x) - p'(x) = x^{2} + 2x + 1$ . Compute $p(5)$ .

Your answer

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

Solution

Solution:
Observe that $p(x)$ must be quadratic. Let $p(x) = a x^{2} + b x + c$ .

Comparing coefficients gives $a = 1$ , $b - 2a = 2$ , and $c - b = 1$ .

So $b = 4$ , $c = 5$ , $p(x) = x^{2} + 4x + 5$ and $p(5) = 25 + 20 + 5 = 50$ .

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