2014 AMC 12B Problems/Problem 16

Problem

Let $P$ be a cubic polynomial with $P(0) = k$ , $P(1) = 2k$ , and $P(-1) = 3k$ . What is $P(2) + P(-2)$ ?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D)}}\ 7k\qquad\textbf{(E)}\ 14k$ (Error compiling LaTeX. Unknown error_msg)

Solution

Let $P(x) = Ax^3+Bx^2 + Cx+D$ . Plugging in $0$ for $x$ , we find $D=k$ , and plugging in $1$ and $-1$ for $x$ , we obtain the following equations: $A+B+C+k=2k$ $-A+B-C+k=3k$ Adding these two equations together, we get $2B=3k$ If we plug in $2$ and $-2$ in for $x$ , we find that $P(2)+P(-2) = 8A+4B+2C+k+(-8A+4B-2C+k)=8B+2k$ Multiplying the third equation by $4$ and adding $2k$ gives us our desired result, so $P(2)+P(-2)=12k+2k=\boxed{\textbf{(E)}\ 14k}$

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2014 AMC 12B Problems/Problem 16

Problem

Solution