Question

How can I write a polynomial equation with the x-intercepts -3,1,3 with all odd multiplicities, and with a y-intercept of -1?

Answer 1

I think you're in the right direction. In order to have an intercept at x=1, your factor needs to be 0, ie., you need (x-1) not (x+1) like you wrote.

So the first attempt gives: f(x) = (x+3)(x-1)(x-3). That gives us the exact right x intercepts. Now what about the y intercept?

Right now it is at f(0) = (0+3)(0-1)(0-3) = +9, that's not good!

So lets multiply bij -1/9 and turn that 9 into a -1:

f(x) = -1/9(x+3)(x-1)(x-3)

Multiplication scales the graph, it doesn't shift it, so the x intercepts stay put. I think that's your answer.

Try pasting that formula in the brilliant Wolfram Alpha website, you'll be amazed!