4.2k views
3 votes
Find a degree 3 polynomial with real coefficients having zeros 5 and 4i and a lead coefficient of 1. Write P in expanded form

1 Answer

3 votes

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.

User Maxadorable
by
8.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories