1999 AMC 12 Problem 17

Attempt Problem 17 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.

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17.

Let

P(x)

be a polynomial such that when

P(x)

is divided by

x - 19,

the remainder is

99,

and when

P(x)

is divided by

x - 99,

the remainder is

19.

What is the remainder when

P(x)

is divided by

(x - 19)(x - 99)?

-x + 80

x + 80

-x + 118

x + 118

0

Dividing by a quadratic leaves a remainder of the form

ax + b

Use

P(19) = 99

and

P(99) = 19

By the Remainder Theorem,

P(19) = 99

and

P(99) = 19.

Write

\begin{aligned} &P(x) = (x - 19)(x - 99)Q(x) \\ &\quad {}+ ax + b. \end{aligned}

Then

19a + b = 99, \qquad 99a + b = 19.

Subtracting gives

80a = -80,

a = -1

and

b = 118.

The remainder is

-x + 118.

Thus, the correct answer is C.

Problem 17 in other years