The function h is defined by \(\mathrm{h(x) = (x - 8)^2(x + 4)(x - 2)}\). The value of \(\mathrm{h(3m -...

1. TRANSLATE the given information

• Given: \(\mathrm{h(x) = (x - 8)^2(x + 4)(x - 2)}\)
• Condition: \(\mathrm{h(3m - 1) = 0}\)
• Find: Sum of all possible values of m

2. TRANSLATE the condition into an equation

• Substitute \(\mathrm{x = 3m - 1}\) into the function:
  \(\mathrm{h(3m - 1) = ((3m - 1) - 8)^2((3m - 1) + 4)((3m - 1) - 2)}\)

3. SIMPLIFY each factor

• First factor: \(\mathrm{(3m - 1) - 8 = 3m - 9}\)
• Second factor: \(\mathrm{(3m - 1) + 4 = 3m + 3}\)
• Third factor: \(\mathrm{(3m - 1) - 2 = 3m - 3}\)
• So: \(\mathrm{h(3m - 1) = (3m - 9)^2(3m + 3)(3m - 3) = 0}\)

4. INFER the solution approach

• Since we have a product equal to zero, use the Zero Product Property
• For the product to equal zero, at least one factor must equal zero
• This gives us three equations to solve

5. SIMPLIFY each equation

• From \(\mathrm{(3m - 9)^2 = 0}\): \(\mathrm{3m - 9 = 0}\) → \(\mathrm{m = 3}\)
• From \(\mathrm{(3m + 3) = 0}\): \(\mathrm{3m = -3}\) → \(\mathrm{m = -1}\)
• From \(\mathrm{(3m - 3) = 0}\): \(\mathrm{3m = 3}\) → \(\mathrm{m = 1}\)

6. SIMPLIFY to find the final answer

• All possible values of m: 3, -1, 1
• Sum: \(\mathrm{3 + (-1) + 1 = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly substitute \(\mathrm{3m - 1}\) into the function, making errors like forgetting to distribute the negative sign or incorrectly combining terms when simplifying expressions like \(\mathrm{(3m - 1) - 8}\).

For example, they might write \(\mathrm{(3m - 1) - 8 = 3m - 1 - 8 = 3m - 7}\) instead of \(\mathrm{3m - 9}\). This leads to wrong factors and ultimately incorrect values of m, causing confusion and potentially guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the Zero Product Property but make algebraic errors when solving the linear equations, such as \(\mathrm{3m - 9 = 0}\) giving \(\mathrm{m = -3}\) instead of \(\mathrm{m = 3}\), or forgetting to solve one of the three equations entirely.

This leads to an incorrect sum because they're adding the wrong m-values or missing some values completely.

The Bottom Line:

This problem requires careful substitution and systematic application of the Zero Product Property. The key challenge is maintaining accuracy through multiple algebraic steps while ensuring all possible solutions are found and correctly summed.