The function g is defined by g(x) = x(x − 2)(x + 6)2.The value of g(7 − w) is 0, where w is a constant.What is the sum of all possible values of w ?
Question
Solution
To find the sum of all possible values of w, we first need to set the equation g(7-w) = 0 and solve for w.
The function g is defined as g(x) = x(x - 2)(x + 6)^2.
So, we substitute (7-w) for x in the equation:
g(7-w) = (7-w)((7-w) - 2)((7-w) + 6)^2 = 0
This simplifies to:
(7-w)(5-w)(13-w)^2 = 0
Setting each factor equal to zero gives the solutions for w:
7 - w = 0 => w = 7 5 - w = 0 => w = 5 13 - w = 0 => w = 13
So, the sum of all possible values of w is 7 + 5 + 13 = 25.
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