1. ## Drawing a polynomial

My question gives a polynomal with a degree of 16:

y = 1/288 (x-3)^2 (x+2)^3 (x+1)^9 (x-4)^2

So the polynomial crosses the x axis at point:
3 two times
-2 three times
-1 nine times
4 two times

My question is how the graph would look like. I guessed that since there really is only 4 x-intercepts and that the rest just overlap, that it's a quartic function, therefore giving the shape M.

But then it's a 16 degree polynomial so it doesnt make sense, is it a combination of quartics and cubics? Can someone help?

2. Originally Posted by onenameless
My question gives a polynomal with a degree of 16:

y = 1/288 (x-3)^2 (x+2)^3 (x+1)^9 (x-4)^2
So the polynomial crosses the x axis at point:
3 two times
-2 three times
-1 nine times
4 two times

My question is how the graph would look like. I guessed that since there really is only 4 x-intercepts and that the rest just overlap, that it's a quartic function, therefore giving the shape M.

But then it's a 16 degree polynomial so it doesnt make sense, is it a combination of quartics and cubics? Can someone help?
When we have repeated root in a polynomial a few different things can happen.

If the root is repeated an even number of time like 3 above the graph bounces (doesn't cross the x-axis) there.

If the root is repeated an odd number of times it will cross the x-axis there, and have a horizontal tangent line at the point. like -2 above is repeated 3 times

The graph will have "U" like end behavior becuase the polynomial has an even degree.

3. oh i see. so i was kind of on the right track, thanks.

4. ## Why do even multiplicities touch but odd pass through?

I am curious as to why roots repeated an even number of times "touch" the x-axis but those repeated an odd number of times pass through? I know using calculus that if a root has multiplicity 2 or higher then that root is also a root of the derivative.