2003 AMC 12A Problem 21

Problem 21 of 25HarderAlgebra

The graph of the polynomial

\begin{aligned} &P(x) = x^5 + ax^4 + bx^3 \\ &\quad {}+ cx^2 + dx + e \end{aligned}

has five distinct

x

-intercepts, one of which is at

(0, 0).

Which of the following coefficients cannot be zero?

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Solution

Since

(0,0)

is an intercept,

P(0)=e=0,

P(x)

=x\left(x^4+ax^3+bx^2+cx+d\right).

The four remaining intercepts are nonzero and distinct, and

d

equals their product, which is therefore nonzero. Any of

a,b,c

can be zero for suitable choices of those roots, but

d\neq0.

Thus, the correct answer is D.