1962 AMC 12 Problem 9

Attempt Problem 9 of the 1962 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 1962 AMC 12 solutions, or check the answer key.

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

When

x^9-x

is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:

more than

5

5

4

3

2

Begin with

x(x^8-1)

and repeatedly use differences of squares

Over the integers, stop after isolating

x,

x-1,

x+1,

x^2+1,

and

x^4+1

Factoring over the integers gives

\begin{aligned} x^9-x &=x(x-1)(x+1)\\ &\quad\cdot(x^2+1)(x^4+1). \end{aligned}

The last two nonconstant factors are irreducible over the integers, so there are

5

factors.

Thus, the correct answer is B.

Problem 9 in other years