Question

find a polynomial function with least degree having the following roots: 3, i, -i

f(x)=x^3 - 3x^2 + x - 3

f(x)=x^3 + 3x^2 + x + 3

f(x)=x^3 - 3x^2 - x + 3

f(x)=x^2 - 4x + 3

Explanation:

Step1: Recall factor - root relationship

If \(r\) is a root of a polynomial, then \((x - r)\) is a factor. Given roots \(3\), \(i\), and \(-i\), the factors are \((x - 3)\), \((x - i)\), and \((x + i)\).

Step2: Multiply the factors

First, multiply \((x - i)\) and \((x + i)\):
\[

$$\begin{align*} (x - i)(x + i)&=x^{2}-i^{2}\\ &=x^{2}+ 1 \end{align*}$$

\]
Then multiply \((x^{2}+1)\) by \((x - 3)\):
\[

$$\begin{align*} (x^{2}+1)(x - 3)&=x^{2}(x - 3)+1\times(x - 3)\\ &=x^{3}-3x^{2}+x - 3 \end{align*}$$

\]

Answer:

\(f(x)=x^{3}-3x^{2}+x - 3\)