Difference between revisions of "2010 AMC 12A Problems/Problem 21"
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− | == Problem | + | == Problem == |
The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the line intersect. What is the largest of these values? | The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the line intersect. What is the largest of these values? | ||
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Suppose we let <math>p</math>, <math>q</math>, and <math>r</math> be the roots of this function, and let <math>x^3-ux^2+vx-w</math> be the cubic polynomial with roots <math>p</math>, <math>q</math>, and <math>r</math>. | Suppose we let <math>p</math>, <math>q</math>, and <math>r</math> be the roots of this function, and let <math>x^3-ux^2+vx-w</math> be the cubic polynomial with roots <math>p</math>, <math>q</math>, and <math>r</math>. | ||
− | + | <cmath>\begin{align*}(x-p)(x-q)(x-r) &= x^3-ux^2+vx-w\ | |
− | < | + | (x-p)^2(x-q)^2(x-r)^2 &= x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0\ |
− | + | \sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} &= x^3-ux^2+vx-w = 0\end{align*}</cmath> | |
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In order to find <math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}</math> we must first expand out the terms of <math>(x^3-ux^2+vx-w)^2</math>. | In order to find <math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}</math> we must first expand out the terms of <math>(x^3-ux^2+vx-w)^2</math>. | ||
− | + | <cmath>(x^3-ux^2+vx-w)^2</cmath> | |
− | < | ||
<math>= x^6-2ux^5+(u^2+2v)x^4-(2uv+2w)x^3+(2uw+v^2)x^2-2vwx+w^2</math> | <math>= x^6-2ux^5+(u^2+2v)x^4-(2uv+2w)x^3+(2uw+v^2)x^2-2vwx+w^2</math> | ||
[Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the last 3 terms of the expansion.] | [Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the last 3 terms of the expansion.] | ||
− | < | + | <cmath>\begin{align*}&2u = 10\ |
+ | u^2+2v &= 29\ | ||
+ | 2uv+2w &= 4\ | ||
+ | u &= 5\ | ||
+ | v &= 2\ | ||
+ | w &= -8\ | ||
+ | &\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} = x^3-5x^2+2x+8\end{align*}</cmath> | ||
− | <math> | + | All that's left is to find the largest root of <math>x^3-5x^2+2x+8</math>. |
− | < | + | <cmath>\begin{align*}&x^3-5x^2+2x+8 = (x-4)(x-2)(x+1)\ |
− | + | &\boxed{\textbf{(A)}\ 4}\end{align*}</cmath> | |
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− | + | == See also == | |
+ | {{AMC12 box|year=2010|num-b=20|num-a=22|ab=A}} | ||
− | + | [[Category:Intermediate Algebra Problems]] |
Revision as of 22:34, 25 February 2010
Problem
The graph of lies above the line except at three values of , where the graph and the line intersect. What is the largest of these values?
Solution
The values in which intersect at are the same as the zeros of .
Since there are zeros and the function is never negative, all zeros must be double roots because the function's degree is .
Suppose we let , , and be the roots of this function, and let be the cubic polynomial with roots , , and .
In order to find we must first expand out the terms of .
[Quick note: Since we don't know , , and , we really don't even need the last 3 terms of the expansion.]
All that's left is to find the largest root of .
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 | |
All AMC 12 Problems and Solutions |