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Advanced Math Difficulty: Hard

The function $g$ is defined by $g (x) = x (x - 2) {(x + 6)}^{2}$ . The value of $g left parenthesis 7 minus w right parenthesis$ is $0$ , where $w$ is a constant. What is the sum of all possible values of $w$ ?

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Explanation

The correct answer is $25$ . The value of $g left parenthesis 7 minus w right parenthesis$ is the value of $g (x)$ when $x = 7 - w$ , where $w$ is a constant. Substituting $7 minus w$ 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 $g left parenthesis 7 minus w right parenthesis$ is $0$ . Substituting $0$ for $g left parenthesis 7 minus w right parenthesis$ 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 $7 equals w$ . 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 $5 equals w$ . 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 $13 equals w$ . 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 $25$ .