Difference between revisions of "1999 AHSME Problems/Problem 12"
m (→Solution) |
m (→Solution added) |
||
Line 7: | Line 7: | ||
==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) = 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. Thus, the answer is <math>\boxed{C}</math>. | + | 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== |
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.