Difference between revisions of "1999 AHSME Problems/Problem 12"
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==Problem== | ==Problem== | ||
− | What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions <math> y | + | What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions <math> y=p(x)</math> and <math> y=q(x)</math>, each with leading coefficient 1? |
<math> \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8</math> | <math> \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8</math> | ||
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==Solution== | ==Solution== | ||
− | + | The intersections of the two polynomials, <math>p(x)</math> and <math>q(x)</math>, are precisely the roots of the equation <math>p(x)=q(x) \rightarrow p(x) - q(x) = 0</math>. Since the leading coefficients of both polynomials are <math>1</math>, the degree of <math>p(x) - q(x) = 0</math> 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 <math>\boxed{C}</math>. | |
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==See Also== | ==See Also== |
Latest revision as of 19:50, 6 March 2016
Problem
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions and
, each with leading coefficient 1?
Solution
The intersections of the two polynomials, and
, are precisely the roots of the equation
. Since the leading coefficients of both polynomials are
, the degree of
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
.
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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