1988 AMC 12 Problem 15

Attempt Problem 15 of the 1988 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 1988 AMC 12 solutions, or check the answer key.

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15.

a

and

b

are integers such that

x^2-x-1

is a factor of

ax^3+bx^2+1,

then

b

-2

-1

0

1

2

Reduce powers using

x^2\equiv x+1

A polynomial of degree below

2

divisible by the quadratic must be zero

Modulo

x^2-x-1,

we have

x^2\equiv x+1

and

x^3\equiv2x+1.

Thus the polynomial is congruent to

(2a+b)x+(a+b+1).

Both coefficients must vanish, so

2a+b=0

and

a+b=-1.

Hence

a=1

and

b=-2.

Thus the correct answer is A.

Problem 15 in other years