Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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 [tex](x-5),(x-4i),(x+4i)[/tex].
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
[tex]P(x)=1(x-5)(x-4i)(x+4i)[/tex]
[tex]P(x)=(x-5)(x^2-(4i)^2)[/tex]
[tex]P(x)=(x-5)(x^2+16)[/tex] [tex][\because i^2=-1][/tex]
[tex]P(x)=(x-5)(x^2+16)[/tex]
On further simplification, we get
[tex]P(x)=x^3+16x-5x^2-80[/tex]
[tex]P(x)=x^3-5x^2+16x-80[/tex]
Therefore, the required polynomial P is [tex]P(x)=x^3-5x^2+16x-80[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
