Question

Determine the x intercepts of the polynomial function:g(x)=x^3+2x^2-9x-18Recall that the x intercept is a point (x,0) where our value for y is always zero. List the intercepts (x,0) from smallest to largest and separate them with a comma between each point.Answer:

Answer 1

We will have the following:

We determine the x-intercepts as follows:

First, we factor the expression; that is:

`x^3+2x^2-9x-18=(x^3+2x^2)+(-9x-18)`
`=x^2(x+2)-9(x+2)=(x+2)(x^2-9)`
`=(x+2)(x-3)(x+3)`

Now, we equal the expression to 0, that is:

`(x+2)(x-3)(x+3)=0\Rightarrow\begin{cases}x+2=0 \\ \\ x-3=0 \\ \\ x+3=0\end{cases}`
`\Rightarrow\begin{cases}x=-2 \\ \\ x=3 \\ \\ x=-3\end{cases}`

So, the x-intercepts are located at:

`(-2,0)`
`(-3,0)`
`(3,0)`

We can see it as follows: