Question

find a polynomial function of least degree having the following roots: 2, 1, - 1
f(x)=x^{3}+2x^{2}+x + 2
f(x)=x^{3}-2x^{2}-x + 2
f(x)=x^{3}-3x - 2
f(x)=x^{3}+2x^{2}+x - 2

Explanation:

Step1: Use root - factor relationship

If \(r\) is a root of a polynomial, then \((x - r)\) is a factor. For roots \(2\), \(1\), and \(- 1\), the factors are \((x - 2)\), \((x - 1)\), and \((x+1)\).

Step2: Multiply the factors

\[

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

\]

Answer:

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