2010 AMC 12A Problem 21

Question

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?

4

5

6

7

8

The graph minus the line is nonnegative with three double roots, so it equals

\big((x-p)(x-q)(x-r)\big)^2

Match the coefficients of

x^5,x^4,x^3

to determine the cubic, then factor it

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.

Problem 21 in other years