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Difference between revisions of "1999 AHSME Problems/Problem 12"

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This is also equivalent to the number of roots of <math>p(x) - q(x) = 0</math>.  Since <math>p(x)</math> and <math>q(x)</math> are both fourth degree polynomials with a leading term of <math>x^4</math>, the <math>x^4</math> 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 <math>3</math> real solutions, leading to an answer of <math>\boxed{C}</math>.
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Since the two graphs are fourth degree polynomials, then, by they can have at most <math>4</math> intersections (real solutions), leading to an answer of <math>\boxed{D}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 19:36, 13 January 2015

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.


Since the two graphs are fourth degree polynomials, then, by they can have at most $4$ intersections (real solutions), leading to an answer of $\boxed{D}$.

See Also

1999 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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All AHSME Problems and Solutions

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