2023 AMC 12B Problem 6

Problem 6 of 25EasierAlgebra Number Theory

When the roots of the polynomial

\begin{gathered} P(x)=(x-1)^1(x-2)^2 \\ {}\cdot (x-3)^3\cdots \\ {}\cdot (x-10)^{10} \end{gathered}

are removed from the number line, what remains is the union of

11

disjoint open intervals. On how many of these intervals is

P(x)

positive?

Show solution

Solution

The exponent of the factor

(x-k)

k,

so the sign of

P

changes at

x=k

only when

k

is odd, i.e. at

1,3,5,7,9.

For

x\gt 10

every factor is positive, so

P\gt 0.

Sweeping left and flipping at each odd root, the positive intervals are

(10,\infty),

(9,10),

(6,7),

(5,6),

(2,3),

and

(1,2)

— six intervals in all. Thus, the correct answer is C.