HMMT February

Algebra Difficulty 5.0 AIME Find the answer

Let $P$ be a polynomial such that $P(x)=P(0)+P(1) x+P(2) x^{2}$ and $P(-1)=1$ . Compute $P(3)$ .

Your answer

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

Solution

Plugging in $x=-1,1,2$ results in the trio of equations $1=P(-1)=P(0)-P(1)+P(2)$ , $P(1)=P(0)+P(1)+P(2) \Rightarrow P(1)+P(2)=0$ , and $P(2)=P(0)+2 P(1)+4 P(2)$ . Solving these as a system of equations in $P(0), P(1), P(2)$ gives $P(0)=-1, P(1)=-1, P(2)=1$ . Consequently, $P(x)=x^{2}-x-1 \Rightarrow P(3)=5$ .

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