Question:The function f is defined by \(\mathrm{f(x) = (x - 3)(x + 1)(x - 2)}\).What is the sum of the...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = (x - 3)(x + 1)(x - 2)}\)
• Need to find: Sum of x-coordinates of x-intercepts

2. TRANSLATE what x-intercepts mean mathematically

• X-intercepts occur where the graph crosses the x-axis
• This happens when \(\mathrm{y = f(x) = 0}\)
• So we need to solve: \(\mathrm{(x - 3)(x + 1)(x - 2) = 0}\)

3. INFER the solution strategy

• When a product equals zero, at least one factor must equal zero
• Set each factor equal to zero and solve separately

4. SIMPLIFY by solving each equation

• From \(\mathrm{x - 3 = 0}\): \(\mathrm{x = 3}\)
• From \(\mathrm{x + 1 = 0}\): \(\mathrm{x = -1}\)
• From \(\mathrm{x - 2 = 0}\): \(\mathrm{x = 2}\)

5. SIMPLIFY the final calculation

• \(\mathrm{Sum = 3 + (-1) + 2 = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding that "x-intercepts" means setting the function equal to zero.

Some students might think they need to expand the polynomial first, or they might confuse x-intercepts with y-intercepts. Without recognizing that x-intercepts occur when \(\mathrm{f(x) = 0}\), they can't begin the solution systematically and resort to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when solving \(\mathrm{x + 1 = 0}\) or arithmetic mistakes in the final sum.

Students might correctly set up the equation but solve \(\mathrm{x + 1 = 0}\) as \(\mathrm{x = 1}\) instead of \(\mathrm{x = -1}\), or they might add \(\mathrm{3 + 1 + 2 = 6}\) by forgetting the negative sign. This leads them to get an incorrect sum.

The Bottom Line:

This problem tests whether students can connect the graphical concept of x-intercepts to the algebraic condition \(\mathrm{f(x) = 0}\), then systematically apply the zero product property. The factored form makes the algebra straightforward once the setup is correct.