2010 AMC 12A Problem 21

Problem 21 of 25HarderAlgebra

The graph of

y=x^6-10x^5

+29x^4-4x^3+ax^2

lies above the line

y=bx+c

except at three values of

x,

where the graph and the line intersect. What is the largest of those values?

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Solution

Let

f(x)

be the graph minus the line. It is nonnegative and vanishes at three points, each a double root, so

f(x)=\big(x^3-Ax^2+Bx-C\big)^2.

Matching coefficients gives

-2A=-10\Rightarrow A=5,

then

A^2+2B=29\Rightarrow B=2,

then

-2C-2AB=-4\Rightarrow C=-8.

Thus the cubic is

x^3-5x^2+2x+8

=(x+1)(x-2)(x-4),

with roots

-1,2,

and

4.

The largest is

4.

Thus, A is the correct answer.

Tagged: polynomial · factoring · completing the square