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== | ||
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| + | 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>. | ||
==See Also== | ==See Also== | ||
{{AHSME box|year=1999|num-b=11|num-a=13}} | {{AHSME box|year=1999|num-b=11|num-a=13}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:50, 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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
