Find a degree 3 polynomial with real coefficients having zeros 5 and 4i and a lead coefficient of 1. Write P in expanded form

Given:

The degree of the polynomial = 3

Leading coefficient = 1

Zeros of the polynomial are 5 and 4i.

To find:

The expanded form of the polynomial.

Solution:

According to the complex conjugate root theorem, if a+ib is a zero of a polynomial, then a-ib is also a zero of that polynomial.

Here, zeros of the polynomial are 5 and 4i. It means the third zero of the polynomial is -4i. So, the factors of the polynomial are .

The required polynomial is the product of its all factor and a constant which is equal to the leading coefficient. Here, the constant is 1. So, the required polynomial is

On further simplification, we get

Therefore, the required polynomial P is .

step-by-step explanation:

there is a formula i can give you:

it is r^2*n*sin(360/n) divided by 2 where r represents the "radius" and n is the number of sides.

plug them in:

(10sqrt(3))^2*6*sin(360/6) divided by 2

100*3*6*sin(60) divided by 2

900*sin(60)

900*sqrt(3)/2

450*sqrt(3)

approx is 779.4 cm^2