Question

Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 3, -2is the only other zero, leading coefficient is 5.

Answer 1

By the given properties of the polynomial

The polynomial is of fifth-degree

This implies that the highest power of the variable is 5

Also, we are given that

2 is a zero of multiplicity 3,

For 2 to be a zero of a polynomial then the expression

`x-2`

is a factor of the polynomial

Since the multiplicity is 3 then the expression becomes

`(x-2)^3`

Also, -2 is the only other zero

This implies,

`x+2`

is a factor of the polynomial

For the polynomial to be of fifth-degree then the factor x + 2 becomes

`(x+2)^2`

The leading coefficient of the polynomial is 5

Therefore, the polynomial is

`y=5(x-2)^3(x+2)^2`