Question:What is an x-coordinate of an x-intercept of the graph of \(\mathrm{y = 5(x - 6)(x + 4)(x + 8)}\)...

1. TRANSLATE the problem requirement

• We need an x-coordinate of an x-intercept
• X-intercepts occur where the graph crosses the x-axis
• Key translation: At x-intercepts, \(\mathrm{y = 0}\)

2. TRANSLATE the given equation into our working equation

• Given: \(\mathrm{y = 5(x - 6)(x + 4)(x + 8)}\)
• Set \(\mathrm{y = 0}\): \(\mathrm{5(x - 6)(x + 4)(x + 8) = 0}\)

3. INFER the solution strategy

• Since we have a product equal to zero, we can use zero product property
• Notice that \(\mathrm{5 ≠ 0}\), so the constant factor doesn't matter
• We need: \(\mathrm{(x - 6)(x + 4)(x + 8) = 0}\)

4. SIMPLIFY by applying zero product property

• For the product to equal zero, at least one factor must equal zero:
  • \(\mathrm{x - 6 = 0}\) OR \(\mathrm{x + 4 = 0}\) OR \(\mathrm{x + 8 = 0}\)
• Solve each equation:
  • \(\mathrm{x - 6 = 0}\) → \(\mathrm{x = 6}\)
  • \(\mathrm{x + 4 = 0}\) → \(\mathrm{x = -4}\)
  • \(\mathrm{x + 8 = 0}\) → \(\mathrm{x = -8}\)

Answer: Any of 6, -4, or -8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might not connect "x-intercept" with the condition \(\mathrm{y = 0}\). They may try to work directly with the factored form without setting up the equation properly, leading to confusion about what to solve for. This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students may correctly set \(\mathrm{y = 0}\) but make sign errors when solving the linear equations. For example, solving \(\mathrm{x + 4 = 0}\) as \(\mathrm{x = 4}\) instead of \(\mathrm{x = -4}\), or \(\mathrm{x - 6 = 0}\) as \(\mathrm{x = -6}\) instead of \(\mathrm{x = 6}\). This leads them to potentially select wrong answer choices or mix up positive and negative values.

The Bottom Line:

The beauty of this problem is that when a polynomial is given in factored form, finding x-intercepts becomes straightforward - but only if you recognize that x-intercepts mean \(\mathrm{y = 0}\) and know how to apply the zero product property correctly.