Question

find a polynomial function of least degree having the following roots: 5, 3, 1
f(x)=x^{3}-9x^{2}+23x - 15
f(x)=x^{3}+9x^{2}+23x + 15
f(x)=x^{3}+23x - 15
f(x)=2x^{3}-9x^{2}-23x + 15

Explanation:

Step1: Use factor - root relationship

If \(r\) is a root of a polynomial, then \((x - r)\) is a factor. For roots \(5\), \(3\), and \(1\), the factors are \((x - 5)\), \((x - 3)\), and \((x - 1)\).

Step2: Multiply the factors

\[

$$\begin{align*} f(x)&=(x - 5)(x - 3)(x - 1)\\ &=(x^{2}-3x-5x + 15)(x - 1)\\ &=(x^{2}-8x + 15)(x - 1)\\ &=x^{3}-x^{2}-8x^{2}+8x+15x - 15\\ &=x^{3}-9x^{2}+23x - 15 \end{align*}$$

\]

Answer:

A. \(f(x)=x^{3}-9x^{2}+23x - 15\)