Question

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

Answer 1

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
`(x-5),(x-4i),(x+4i)`.

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

`P(x)=1(x-5)(x-4i)(x+4i)`

`P(x)=(x-5)(x^2-(4i)^2)`

`P(x)=(x-5)(x^2+16)`
`[\because i^2=-1]`

`P(x)=(x-5)(x^2+16)`

On further simplification, we get

`P(x)=x^3+16x-5x^2-80`

`P(x)=x^3-5x^2+16x-80`

Therefore, the required polynomial P is
`P(x)=x^3-5x^2+16x-80`.