1986 AMC 12 Problem 24

Attempt Problem 24 of the 1986 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 1986 AMC 12 solutions, or check the answer key.

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24.

Let

p(x)=x^2+bx+c,

where

b

and

c

are integers. If

p(x)

is a factor of both

x^4+6x^2+25

and

3x^4+4x^2+28x+5,

what is

p(1)?

0

1

2

4

8

A common factor divides every integer linear combination of the two polynomials

Subtract the second polynomial from three times the first

The common factor

p(x)

divides

\begin{aligned} &3(x^4+6x^2+25)\\ &\quad{}-(3x^4+4x^2+28x+5)\\ &\qquad=14(x^2-2x+5). \end{aligned}

Because

p(x)

is monic with integer coefficients, Gauss’s lemma implies that it divides

x^2-2x+5.

The two polynomials are monic and have the same degree, so

p(x)=x^2-2x+5.

Hence

p(1)=1-2+5=4.

Thus the correct answer is D.

Problem 24 in other years