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 \equal{} p(x)</math> and <math> y \equal{} q(x)</math>, each with leading coefficient 1?
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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==
  
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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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>.
 
 
 
 
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}}
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{{MAA Notice}}

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 $y=p(x)$ and $y=q(x)$, 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

The intersections of the two polynomials, $p(x)$ and $q(x)$, are precisely the roots of the equation $p(x)=q(x) \rightarrow p(x) - q(x) = 0$. Since the leading coefficients of both polynomials are $1$, the degree of $p(x) - q(x) = 0$ 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 $\boxed{C}$.

See Also

1999 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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