Find a formula for P(x).

Question

The polynomial of degree 4, P(x) has a root of multiplicity of 2 at x=2 and roots of multiplicity 1 at x=0 and x= -2. It goes through the point (5,220.5)

P(x)= ?

Answer 1

If 2 is a root, then (x-2) is a factor. The multiplicity is 2 so it is a factor twice, i.e. (x-2)²

If 0 is a root, then x is a factor (x-0).

If -2 is a root, then [x-(-2)] = (x+2) is a factor.

So far we have:

P(x) = Ax(x-2)²(x+2)

The roots will be the same regardless of the value of A (some real number).

But since the function passes through (5, 220.5) that info can be used to find A.

220.5 = A(5)(3)²(7)

A= 7/10

P(x) = 7/10 (x²+2x)(x²-4x+4)

The following graph shows this along with two different flavors for the P(x).

desmos.com/calculator/d4gqvetqa2