Difference between revisions of "1999 AHSME Problems/Problem 12"
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==Solution== | ==Solution== | ||
| − | { | + | Finding the number of solutions to <math>p(x) = q(x)</math> will find the number of intersections of the two graphs. |
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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>. | ||
==See Also== | ==See Also== | ||
{{AHSME box|year=1999|num-b=11|num-a=13}} | {{AHSME box|year=1999|num-b=11|num-a=13}} | ||
Revision as of 11:08, 8 August 2011
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?
Solution
Finding the number of solutions to 
will find the number of intersections of the two graphs.
This is also equivalent to the number of roots of 
. Since 
and 
are both fourth degree polynomials with a leading term of 
, the 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 
real solutions, leading to an answer of 
.
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 | ||
