Question

MEDAL AND FAN!

Prove y =((x - 1)^2 (x - 3)(9 - x)^2)/((x+2)(x - 5)^2 (9 - x)) fits the requirements below (without using a graph to prove it).

It has a vertical asymptote at x = -2; as x approaches -2 from the left f(x) approaches positive infinity, and as x approaches -2 from the right f(x) approaches negative infinity.

It has a vertical asymptote at x = 5; as x approaches 5 from both the left and the right, f(x) approaches positive infinity.

It is undefined at x = 9 but does not have an asymptote there.

It is zero at x = 1 and x = 3.

It is positive when $x$ is in any of the intervals (-infinity, -2), (3, 5) and (5, 9).

It is negative when x is in any of the following intervals (-infinity, -2), (3, 5) and (5, 9). It is negative when x is in any of the intervals (-2, 1), (1, 3), and (9, +infinity).

Please help.