To determine the [tex]\( x \)[/tex]-intercepts of the polynomial function [tex]\( g(x) = x^3 + 2x^2 - 9x - 18 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex]. This involves solving the equation:
[tex]\[
x^3 + 2x^2 - 9x - 18 = 0
\][/tex]
Here is a step-by-step solution to find these intercepts:
1. Setting up the equation:
[tex]\[
x^3 + 2x^2 - 9x - 18 = 0
\][/tex]
2. Solving the equation:
We can solve the polynomial equation by finding its roots.
The solutions to this equation are:
[tex]\[
x = -3, -2, 3
\][/tex]
3. Formatting the intercepts as points:
We will express the intercepts in the form [tex]\((x, 0)\)[/tex].
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
(-3, 0)
\][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
(-2, 0)
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
(3, 0)
\][/tex]
4. Listing the intercepts:
We list the intercepts from smallest to largest [tex]\( x \)[/tex]-value:
[tex]\[
(-3, 0), (-2, 0), (3, 0)
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts of the polynomial function [tex]\( g(x) = x^3 + 2x^2 - 9x - 18 \)[/tex] are:
[tex]\[
(-3, 0), (-2, 0), (3, 0)
\][/tex]
This is the final answer.
