what is the root of the polynomial equation $x(x - 2)(x + 3)=18$? use a graphing calculator and a system of…

Explanation:

Step1: Expand the left - hand side

First, expand $x(x - 2)(x + 3)$: [ \begin{align*} x(x - 2)(x + 3)&=x(x^{2}+3x-2x - 6)\ &=x(x^{2}+x - 6)\ &=x^{3}+x^{2}-6x \end{align*} ] So the equation becomes $x^{3}+x^{2}-6x=18$, or $x^{3}+x^{2}-6x - 18 = 0$.

Step2: Use the rational root theorem (alternatively, test the given values)

Let's test the given values one by one. For $x=-3$: [ \begin{align*} (-3)^{3}+(-3)^{2}-6\times(-3)-18&=-27 + 9+18-18\ &=-18+18 - 18\ &=-18\neq0 \end{align*} ] For $x = 0$: [ \begin{align*} 0^{3}+0^{2}-6\times0-18&=- 18\neq0 \end{align*} ] For $x = 2$: [ \begin{align*} 2^{3}+2^{2}-6\times2-18&=8 + 4-12-18\ &=12-12-18\ &=-18\neq0 \end{align*} ] For $x = 3$: [ \begin{align*} 3^{3}+3^{2}-6\times3-18&=27+9 - 18-18\ &=36-18-18\ &=0 \end{align*} ]

Answer:

3