Construct a polynomial function with the following properties: fifth degree,-2is a zero of multiplicity 2,2 is the only other zero.

Answer 1

If -2 is a root/zero, then (x+2) must be a factor of the polynomial function. The multiplicity value of 2 means it must appear as a factor twice. So, so far we have f(x) = (x+2)².

Because 2 is the only other zero, (x-2) is a factor and it must appear three times in order for the entire function to be of degree 5.

So,

f(x) = (x+2)²(x-2)³

Here is a graph with the zeroes labelled. It also shows the other turning point of the graph at x = -0.4, but it takes some calculus to determine that value.

desmos.com/calculator/cyhhtft1if