To solve for the value of the polynomial function [tex]\( g(x) \)[/tex] at [tex]\( x = -1 \)[/tex], we will follow these steps:
Given the polynomial [tex]\( g(x) = x^3 + 6x^2 + 12x + 8 \)[/tex],
1. Substitute [tex]\( x = -1 \)[/tex] into the polynomial equation:
[tex]\[
g(-1) = (-1)^3 + 6(-1)^2 + 12(-1) + 8
\][/tex]
2. Compute each term individually:
- Compute [tex]\( (-1)^3 \)[/tex]:
[tex]\[
(-1)^3 = -1
\][/tex]
- Compute [tex]\( 6(-1)^2 \)[/tex]:
[tex]\[
(-1)^2 = 1 \quad \text{so} \quad 6 \cdot 1 = 6
\][/tex]
- Compute [tex]\( 12(-1) \)[/tex]:
[tex]\[
12 \cdot (-1) = -12
\][/tex]
- The constant term is [tex]\( 8 \)[/tex].
3. Add these computed values together:
[tex]\[
g(-1) = -1 + 6 - 12 + 8
\][/tex]
4. Simplify the expression step by step:
- First combine [tex]\(-1\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[
-1 + 6 = 5
\][/tex]
- Then combine [tex]\(5\)[/tex] and [tex]\(-12\)[/tex]:
[tex]\[
5 - 12 = -7
\][/tex]
- Finally, combine [tex]\(-7\)[/tex] and [tex]\(8\)[/tex]:
[tex]\[
-7 + 8 = 1
\][/tex]
Therefore, [tex]\( g(-1) = 1 \)[/tex].
So the value of the function when [tex]\( x = -1 \)[/tex] is [tex]\( g(-1) = 1 \)[/tex].
