Question

Find the​ fourth-degree polynomial function with zeros 4​, -4, 4i ​, and -4i . Write the function in factored form.

Answer 1

Given:

A fourth-degree polynomial function has zeros 4​, -4, 4i ​, and -4i .

To find:

The fourth-degree polynomial  function in factored form.

Solution:

The factor for of nth degree polynomial is:

$P(x)=(x-a_1)(x-a_2)...(x-a_n)$

Where, $a_1,a_2,...,a_n$ are n zeros of the polynomial.

It is given that a fourth-degree polynomial function has zeros 4​, -4, 4i ​, and -4i. So, the factor form of given polynomial is:

$P(x)=(x-4)(x-(-4))(x-4i)(x-(-4i))$

$P(x)=(x-4)(x+4)(x-4i)(x+4i)$

$P(x)=(x-4)(x+4)(x^2-(4i)^2)$           $[\because a^2-b^2=(a-b)(a+b)]$

On further simplification, we get

$P(x)=(x-4)(x+4)(x^2-4^2i^2)$

$P(x)=(x-4)(x+4)(x^2+16)$                $[\because i^2=-1]$

Therefore, the required fourth degree polynomial is $P(x)=(x-4)(x+4)(x^2+16)$.