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 \((x + i)\) using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = x\) and \(b = i\). So \((x - i)(x + i)=x^{2}-i^{2}=x^{2}+1\).
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:

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