1999 AHSME Problems/Problem 12

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Problem

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$, each with leading coefficient 1?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

Solution

The intersections of the two polynomials, $p(x)$ and $q(x)$, are precisely the roots of the equation $p(x)=q(x) \rightarrow p(x) - q(x) = 0$. Since the leading coefficients of both polynomials are $1$, the degree of $p(x) - q(x) = 0$ is at most three, and the maximum point of intersection is three, because a third degree polynomial can have at most three roots. Thus, the answer is $\boxed{C}$.

See Also

1999 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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