OG详解-OG5 数学1 Q27

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 g(7-w) = (7-w)(5-w)(13-w)² 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.