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 $P(x)=(x-4)(x+4)(x^2+16)$.

Answer 2

The fourth degree polynomial is given by f(x) = (x - 4)(x + 4)(x + 4i)(x - 4i)

Polynomial

Polynomial is an expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Polynomials can be classified based on degree as linear, quadratic, cubic, fourth degree.

Let f(x) represent the function with zeros 4​, -4, 4i ​, and -4i. Hence:

x = 4; x = -4; x = 4i; x = -4i

x - 4 = 0; x + 4 = 0; x - 4i = 0; x + 4i = 0

f(x) = (x - 4)(x + 4)(x + 4i)(x - 4i)

The fourth degree polynomial is given by f(x) = (x - 4)(x + 4)(x + 4i)(x - 4i)

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