1969 AMC 12 Problem 34

Attempt Problem 34 of the 1969 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 1969 AMC 12 solutions, or check the answer key.

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

34.

The remainder

R

obtained by dividing

x^{100}

by

x^2-3x+2

is a polynomial of degree less than

2.

Then

R

may be written as:

2^{100}-1

2^{100}(x-1)-(x-2)

2^{100}(x-3)

x(2^{100}-1)+2(2^{99}-1)

2^{100}(x+1)-(x+2)

Answer: B

Concepts:polynomial system of equations substitution

Difficulty rating: 2010

Small hint:

Write

R(x)=ax+b

and use

x^2-3x+2=(x-1)(x-2)

Big hint:

Evaluate the division identity at

x=1

and

x=2

Solution:

Write

R(x)=ax+b.

Since the divisor is

(x-1)(x-2),

evaluating the division identity at its roots gives

R(1)=1,\qquad R(2)=2^{100}.

Therefore

\begin{aligned} R(x)&=1+(2^{100}-1)(x-1)\\ &=2^{100}(x-1)-(x-2). \end{aligned}

Therefore, the correct answer is B.