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 $y \equal{} p(x)$ (Error compiling LaTeX. Unknown error_msg) and $y \equal{} q(x)$ (Error compiling LaTeX. Unknown error_msg), 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

Finding the number of solutions to $p(x) = q(x)$ will find the number of intersections of the two graphs.


This is also equivalent to the number of roots of $p(x) - q(x) = 0$. Since $p(x)$ and $q(x)$ are both fourth degree polynomials with a leading term of $x^4$, the $x^4$ term will drop out, leaving at most a third degree polynomial (cubic) on the left side. By the Fundamental Theorem of Algebra, a cubic polynomial can have at most $3$ real solutions, leading to an answer of $\boxed{C}$.

See Also

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