Question

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?

$a$

$b$

$c$

$d$

$e$

Since $(0,0)$ is an intercept, $P(0)=e=0,$ so $P(x)$ $=x\left(x^4+ax^3+bx^2+cx+d\right).$

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.