2002 AMC 12A Problem 25

Question

The nonzero coefficients of a polynomial

P

with real coefficients are all replaced by their mean to form a polynomial

Q.

Which of the following could be a graph of

y = P(x)

and

y = Q(x)

over the interval

-4 \le x \le 4?

The sum of the coefficients of

P

equals the sum of the coefficients of

Q

Evaluating at

x = 1

gives

P(1) = Q(1),

so the two graphs must intersect at

x = 1

Replacing the nonzero coefficients by their mean keeps the total of the coefficients unchanged, so

P

and

Q

have the same coefficient sum. Since

P(1)

and

Q(1)

each equal that sum,

P(1) = Q(1).

Therefore the graphs of

y = P(x)

and

y = Q(x)

must cross at

x = 1.

The only choice showing an intersection at

x = 1

is graph B. (There,

P(x) = 2x^4 - 3x^2 - 3x - 4

and

Q(x) = -2x^4 - 2x^2 - 2x - 2.

)

Thus, the correct answer is B.

Problem 25 in other years