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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The correct answer is 25. The value of g(7 − w) is the value of g(x) when x = 7 − w, where w is a constant. Substituting 7 − w for x in the given equation yields g(7 − w) = (7 − w)(7 − w − 2)(7 − w + 6) 2 , which is equivalent to g(7 − w) = (7 − w)(5 − w)(13 − w) 2 . It’s given that the value of g(7 − w) is 0. Substituting 0 for g(7 − w) in the equation yields 0 = (7 − w)(5 − w)(13 − w) 2 . Since the product of the three factors on the right-hand side of this equation is equal to 0, at least one of these three factors must be equal to 0. Therefore, the possible values of w can be found by setting each factor equal to 0. Setting the first factor equal to 0 yields 7 − w = 0. Adding w to both sides of this equation yields 7 = w. Therefore, 7 is one possible value of w. Setting the second factor equal to 0 yields 5 − w = 0. Adding w to both sides of this equation yields 5 = w. Therefore, 5 is a second possible value of w. Setting the third factor equal to 0 yields (13 − w) 2 = 0. Taking the square root of both sides of this equation yields 13 − w = 0. Adding w to both sides of this equation yields 13 = w. Therefore, 13 is a third possible value of w. Adding the three possible values of w yields 7 + 5 + 13, or 25. Therefore, the sum of all possible values of w is 25.

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