2003 AMC 12A Problem 21

Attempt Problem 21 of the 2003 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12A solutions, or check the answer key.

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21.

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?

a

b

c

d

e

P(0)=0

forces

e=0

Factor out

x;

the constant term of the remaining quartic is the product of the four nonzero roots

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.

Problem 21 in other years