54.8k views
1 vote
Finding a polynomial of a given degree with given zeros

Finding a polynomial of a given degree with given zeros-example-1

1 Answer

5 votes

Since f(x) is a polynomial with 3rd degree, then it will have 3 roots (zeroes)

One of them is real and the other two are complex conjugate roots

Since the real root is 4, then

x = 4

Since the complex root is (1 - i), then

The other root will be the conjugate of it (1 + i)

x = (1 - i)

x = (1 + i)

To find f(x) we will multiply the three factors of it

We can get the factors from the zeroes


x=4

Subtract 4 from both sides


\begin{gathered} x-4=4-4 \\ x-4=0 \end{gathered}

The first factor is (x - 4)


\begin{gathered} x=1-i \\ x-(1-i)=(1-i)-(1-i) \\ x-1+i=0 \end{gathered}

The second factor is (x - 1 + i)

The third factor is (x - 1 - i)


f(x)=(x-4)(x-1+i)(x-1-i)

We will multiply them to find f(x)


\begin{gathered} (x-1+i)(x-1-i)= \\ x^2-x-ix-x+1+i+ix-i-(i^2)= \\ x^2-2x+1-(-1)= \\ x^2-2x+1+1= \\ x^2-2x+2 \end{gathered}

Multiply it by (x - 4)


\begin{gathered} f(x)=(x-4)(x^2-2x+2) \\ f(x)=x^3-2x^2+2x-4x^2+8x-8 \\ f(x)=x^3-6x^2+10x-8 \end{gathered}

The answer is


f(x)=x^3-6x^2+10x-8

User Mafu Josh
by
8.4k 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