To solve for the sum of all possible values of [tex]\( w \)[/tex] such that [tex]\( g(7-w) = 0 \)[/tex], begin by substituting [tex]\( 7-w \)[/tex] into the function [tex]\( g(x) \)[/tex]:
Given:
[tex]\[ g(x) = x(x - 2)(x + 6)^2 \][/tex]
We substitute [tex]\( x = 7 - w \)[/tex] into the function:
[tex]\[ g(7 - w) = (7 - w)\left((7 - w) - 2\right)\left((7 - w) + 6\right)^2 \][/tex]
This transforms [tex]\( g(7 - w) \)[/tex] into:
[tex]\[ g(7 - w) = (7 - w)((7 - w) - 2)((7 - w) + 6)^2 \][/tex]
Expanding the terms individually:
1. [tex]\((7 - w)\)[/tex]
2. [tex]\((7 - w) - 2 = 5 - w\)[/tex]
3. [tex]\((7 - w) + 6 = 13 - w\)[/tex]
Thus we get:
[tex]\[ g(7 - w) = (7 - w)(5 - w)(13 - w)^2 \][/tex]
We need to solve for [tex]\( w \)[/tex] such that:
[tex]\[ (7 - w)(5 - w)(13 - w)^2 = 0 \][/tex]
This equation will be zero if any of the factors are zero. We set each factor equal to zero:
1. [tex]\( 7 - w = 0 \implies w = 7 \)[/tex]
2. [tex]\( 5 - w = 0 \implies w = 5 \)[/tex]
3. [tex]\( 13 - w = 0 \implies w = 13 \)[/tex]
Hence, the possible values of [tex]\( w \)[/tex] are 7, 5, and 13.
To find the sum of all possible values of [tex]\( w \)[/tex]:
[tex]\[ \text{Sum of } w = 7 + 5 + 13 = 25 \][/tex]
Therefore, the sum of all possible values of [tex]\( w \)[/tex] is [tex]\( \boxed{25} \)[/tex].
