2003 AMC 12A Problems/Problem 21

Problem

The graph of the polynomial

$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$

has five distinct $x$ -intercepts, one of which is at $(0,0)$ . Which of the following coefficients cannot be zero?

$\text{(A)}\ a \qquad \text{(B)}\ b \qquad \text{(C)}\ c \qquad \text{(D)}\ d \qquad \text{(E)}\ e$

Solution

Solution 1

According to Vieta's formulas, the sum of the roots of a 5th degree polynomial taken 4 at a time is $\frac{a_1}{a_5} = d$ . Calling the roots $r_1, r_2, r_3, r_4, r_5$ and letting $r_1 = 0$ (our given zero at the origin), the only way to take four of the roots without taking $r_1$ is $r_2r_3r_4r_5$ . Since all of the other products of 4 roots include $r_1$ , they are all equal to $0$ . And since all of our roots are distinct, none of the terms in $r_2r_3r_4r_5$ can be zero, meaning the entire expression is not zero. Therefore, $d$ is a sum of zeros and a non-zero number, meaning it cannot be zero, so $\mathrm{(D)}$ .

Solution 2

Clearly, since $(0,0)$ is an intercept, $e$ must be $0$ . But if $d$ was $0$ , $x^2$ would divide the polynomial, which means it would have a double root at $0$ , which is impossible, since all five roots are distinct.

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2003 AMC 12A Problems/Problem 21

Problem

Contents

Solution

Solution 1

Solution 2

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