22.7k views
2 votes
Hello, just want to make sure my answer is correct. Thank you!

Hello, just want to make sure my answer is correct. Thank you!-example-1

1 Answer

7 votes

Given the zeros:


\begin{gathered} x_1=4 \\ x_2=4-5i \end{gathered}

You need to remember that the Factor Theorem states that, if, for a polynomial:


f(a)=0

Then, this is a factor of the polynomial:


(x-a)

In this case, you know that:


\begin{gathered} f(4)=0 \\ f(4-5i)=0 \end{gathered}

Therefore, you can determine that these are factors of the polynomial:


\begin{gathered} (x-4) \\ (x-(4-5i)) \end{gathered}

By definition, Complex Conjugates have this form:


(a+bi)(a-bi)

Therefore, you can determine that this is also a factor:


(x-(4+5i))

Now you can set up that the Factored Form of the polynomial is:


(x-4)(x-(4-5i))(x-(4+5i))

You need to expand the expression by applying the Distributive Property and applying:


(a-b)(a+b)=a^2-b^2

Then:


=(x-4)((x-4)^2-25i^2)

By definition:


(a-b)^2=a^2-2ab+b^2

Then, you get:


=(x-4)(x^2-(2)(x)(4)+4^2-25i^2)
=(x-4)(x^2-8x+16-25i^2)

Knowing that:


i^2=-1

And adding the like terms, you get:


=(x-4)(x^2-8x+16-25(-1))
=(x-4)(x^2-8x+16+25)
=(x-4)(x^2-8x+41)

Applying the Distributive Property and adding the like terms, you get:


=(x^2)(x)-(x)(8x)+(x)(41)-(4)(x^2)+(4)(8x)-(4)(41)
=x^3-8x^2+41x-4x^2+32x-164
=x^3-12x^2+73x-164

Hence, the answer is: Second option.

User Andy Gaskell
by
8.7k 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