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

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 $w$ ?

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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.

This problem has been solved

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