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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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Roots", "Function Substitution" ], "current_concepts": [ "Polynomial Roots", "Function Substitution", "Algebraic Equations" ] } ## Find the roots of the function g(x) $$ \begin{aligned} &g(x) = x(x - 2)(x + 6)^2 = 0 \ &x = 0 \quad \text{or} \quad x = 2 \quad \text{or} \quad x = -6 \end{aligned} $$ ## Set up equations for w $$ \begin{aligned} &7 - w = 0 \implies w = 7 \ &7 - w = 2 \implies w = 5 \ &7 - w = -6 \implies w = 13 \end{aligned} $$ ## Calculate the sum of all possible values of w $$ \text{Sum} = 7 + 5 + 13 = 25 $$ 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 $w$? 25 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Roots" ] }

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Find the roots of the function g(x)

Set up equations for w

Calculate the sum of all possible values of w

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