1976 AMC 12 Problem 19

Attempt Problem 19 of the 1976 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 1976 AMC 12 solutions, or check the answer key.

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

19.

A polynomial

p(x)

has remainder three when divided by

x-1

and remainder five when divided by

x-3.

The remainder when

p(x)

is divided by

(x-1)(x-3)

x-2

x+2

2

8

15

The desired remainder has degree less than two, so write it as

ax+b

Use the remainder theorem at

x=1

and

x=3

Let the remainder be

r(x)=ax+b.

Since

p(1)=3

and

p(3)=5,

we have

a+b=3,

and

3a+b=5.

Thus

a=1,

and

b=2,

so the remainder is

x+2.

Therefore, the correct answer is B.

Problem 19 in other years