The function g is defined by \(\mathrm{g(x) = x(x - 2)(x + 6)^2}\). The value of \(\mathrm{g(7 - w)}\) is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = x(x - 2)(x + 6)^2}\)
  • \(\mathrm{g(7 - w) = 0}\) where w is a constant
  • Need to find the sum of all possible values of w
• What this tells us: We need to substitute \(\mathrm{x = 7 - w}\) into the function and set the result equal to zero

2. SIMPLIFY by substituting the expression

• Substitute \(\mathrm{x = 7 - w}\) into \(\mathrm{g(x)}\):
  • \(\mathrm{g(7 - w) = (7 - w)((7 - w) - 2)((7 - w) + 6)^2}\)
  • Simplify each factor:
    • First factor: \(\mathrm{(7 - w)}\)
    • Second factor: \(\mathrm{(7 - w - 2) = (5 - w)}\)
    • Third factor: \(\mathrm{((7 - w) + 6)^2 = (13 - w)^2}\)
  • So \(\mathrm{g(7 - w) = (7 - w)(5 - w)(13 - w)^2}\)

3. INFER the solution strategy using the Zero Product Property

• Since \(\mathrm{g(7 - w) = 0}\), we have: \(\mathrm{(7 - w)(5 - w)(13 - w)^2 = 0}\)
• By the Zero Product Property, at least one factor must equal zero
• This gives us three equations to solve:
  • \(\mathrm{7 - w = 0}\)
  • \(\mathrm{5 - w = 0}\)
  • \(\mathrm{(13 - w)^2 = 0}\)

4. SIMPLIFY to solve each equation

• From \(\mathrm{7 - w = 0}\): \(\mathrm{w = 7}\)
• From \(\mathrm{5 - w = 0}\): \(\mathrm{w = 5}\)
• From \(\mathrm{(13 - w)^2 = 0}\): Taking the square root gives \(\mathrm{13 - w = 0}\), so \(\mathrm{w = 13}\)

5. TRANSLATE to find the final answer

• Sum all possible values of w: \(\mathrm{7 + 5 + 13 = 25}\)

Answer: 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "\(\mathrm{g(7 - w) = 0}\)" means they need to substitute the entire expression \(\mathrm{(7 - w)}\) for x in the original function. Instead, they might try to work with the original function \(\mathrm{g(x)}\) directly or get confused about what substitution to make.

This confusion leads them to get stuck early and abandon systematic solution, often resulting in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the substitution but make algebraic errors when simplifying \(\mathrm{(7 - w - 2)}\) to \(\mathrm{(5 - w)}\) or \(\mathrm{(7 - w + 6)}\) to \(\mathrm{(13 - w)}\), leading to incorrect factors in the final product.

These algebraic mistakes cause them to find incorrect values of w, leading to a wrong sum.

The Bottom Line:

This problem tests whether students can handle function substitution with algebraic expressions and systematically apply the Zero Product Property. The key challenge is managing the multiple algebraic steps while keeping track of the overall goal.