What is an x-coordinate of an x-intercept of the graph of \(\mathrm{y = 3(x - 14)(x + 5)(x + 4)}\)...

1. TRANSLATE the problem information

• Given: \(\mathrm{y = 3(x - 14)(x + 5)(x + 4)}\)
• Need: x-coordinate of an x-intercept
• What this tells us: X-intercepts occur where the graph crosses the x-axis, so \(\mathrm{y = 0}\)

2. INFER the solving approach

• Since the polynomial is already in factored form, we can use the zero product property
• Set the equation equal to zero: \(\mathrm{0 = 3(x - 14)(x + 5)(x + 4)}\)
• The factor of 3 doesn't affect where \(\mathrm{y = 0}\), so we can focus on: \(\mathrm{0 = (x - 14)(x + 5)(x + 4)}\)

3. APPLY the zero product property

• If a product equals zero, at least one factor must equal zero
• This gives us three equations to solve:
  • \(\mathrm{x - 14 = 0}\)
  • \(\mathrm{x + 5 = 0}\)
  • \(\mathrm{x + 4 = 0}\)

4. SIMPLIFY each linear equation

• From \(\mathrm{x - 14 = 0}\): \(\mathrm{x = 14}\)
• From \(\mathrm{x + 5 = 0}\): \(\mathrm{x = -5}\)
• From \(\mathrm{x + 4 = 0}\): \(\mathrm{x = -4}\)

Answer: 14, -5, or -4 (any one of these is correct)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not connect "x-intercept" with the condition \(\mathrm{y = 0}\), instead trying to solve for when \(\mathrm{x = 0}\) or attempting to find the vertex. This fundamental misunderstanding leads to confusion and abandoning systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the zero product property but make sign errors when solving the linear equations, particularly confusing \(\mathrm{x + 5 = 0 \rightarrow x = 5}\) instead of \(\mathrm{x = -5}\). This leads them to provide incorrect x-coordinates like 5 or 4 instead of the correct values.

The Bottom Line:

This problem tests whether students understand what x-intercepts represent and can execute the zero product property correctly. The factored form makes it straightforward once students recognize the connection between intercepts and \(\mathrm{y = 0}\).