Question

hey! how can i find a formula for p(x) with a degree of 4, a root of multiplicity 2 at x=4 and roots of multiplicity 1 at x=0 and x= -4, also it goes through the point (5, 18)?

Answer 1

If you have roots with multiplicity 2 at 4 and multiplicity 1 at x=0 and x=-4 you have something in the form of:
\[ax(x-4)^2(x+4)\]
The constant a SHOULD dictate the point it goes through.
We have:
\[18=a(5)(5-4)^2(5+4)=45a \rightarrow a=18/45\]
So your polynomial is :
\[\frac{18x}{45}(x-4)^2(x+4)\]