2001 AMC 12 Problems/Problem 19

Problem

The polynomial $p(x) = x^3+ax^2+bx+c$ has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The $y$ -intercept of the graph of $y=p(x)$ is 2. What is $b$ ?

$(\mathrm{A})\ -11 \qquad (\mathrm{B})\ -10 \qquad (\mathrm{C})\ -9 \qquad (\mathrm{D})\ 1 \qquad (\mathrm{E})\ 5$

Solution

We are given $c=2$ . So the product of the roots is $-c = -2$ by Vieta's theorem. Vieta's also tells us $\frac{-a}{3}$ is the average of the zeros, so $\frac{-a}3=-2 \implies a = 6$ . We are also given that the sum of the coefficients is $-2$ , so $1+6+b+2 = -2 \implies b=-11$ . So the answer is $\mathrm{A}$ .

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2001 AMC 12 Problems/Problem 19

Problem

Solution