1999 AMC 12 Problem 12

Attempt Problem 12 of the 1999 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

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?

1

2

3

4

8

Intersections occur where

p(x) = q(x)

The leading

x^4

terms cancel in

p(x) - q(x)

The

x

-coordinates of the intersection points are the roots of

p(x) - q(x).

Because both leading coefficients are

1,

the

x^4

terms cancel, so

p(x) - q(x)

has degree at most

3

and therefore at most

3

roots. The bound is attainable, for example by taking

p(x) = x^4

and

q(x) = x^4 - x(x-1)(x+1),

whose graphs intersect at

x = -1, 0, 1.

Thus, the correct answer is C.

Problem 12 in other years