2012/13 AHSMC Part 1 Problem 4

Problem 4 of 16

A quadratic polynomial $f(x)$ satisfies $f(0) = 1$ , $f(1) = 0$ and $f(2) = 3$ . What is the value of $f(3)$ ?

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Solution

Let $f(x) = ax^2 + bx + c$ . Then $1 = f(0) = c$ , $0 = f(1) = a + b + c$ and $3 = f(2) = 4a + 2b + c$ . We have $c = 1$ , $a + b = -1$ and $2a + b = 1$ . Hence $a = 2$ and $b = -3$ , so that $f(3) = 10$ . The answer is (D).

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