\(\mathrm{y = 2(x - d)(x + d)(x + g)(x - d)}\) In the given equation, d and g are unique...

1. TRANSLATE the problem information

• Given: \(\mathrm{y = 2(x - d)(x + d)(x + g)(x - d)}\) where d and g are unique positive constants
• Find: Number of distinct x-intercepts

2. TRANSLATE what "x-intercepts" means mathematically

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

3. INFER the solution strategy using zero product property

• Since we have a product of factors equal to zero, at least one factor must be zero
• The factor 2 can never be zero, so focus on the other factors
• Set each factor equal to zero and solve

4. SIMPLIFY by solving each factor

• From \(\mathrm{(x - d) = 0}\): \(\mathrm{x = d}\)
• From \(\mathrm{(x + d) = 0}\): \(\mathrm{x = -d}\)
• From \(\mathrm{(x + g) = 0}\): \(\mathrm{x = -g}\)

5. INFER the key insight about repeated factors

• Notice that \(\mathrm{(x - d)}\) appears twice in the original expression
• This means \(\mathrm{x = d}\) is a repeated root, but it's still just ONE x-intercept
• Don't count repeated factors as separate intercepts

6. APPLY CONSTRAINTS using the given information

• Since d and g are unique positive constants: \(\mathrm{d ≠ g}\) and both \(\mathrm{d > 0}\), \(\mathrm{g > 0}\)
• This means our three solutions are distinct:
  • \(\mathrm{x = d}\) (positive value)
  • \(\mathrm{x = -d}\) (negative value)
  • \(\mathrm{x = -g}\) (different negative value since \(\mathrm{g ≠ d}\))

Answer: B. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding the effect of repeated factors in polynomial expressions.

Students see \(\mathrm{(x - d)}\) appearing twice and think this creates two separate x-intercepts, counting: \(\mathrm{x = d}\) (first occurrence), \(\mathrm{x = d}\) (second occurrence), \(\mathrm{x = -d}\), and \(\mathrm{x = -g}\) for a total of 4 intercepts. They don't realize that repeated factors represent the same x-intercept with higher multiplicity.

This may lead them to select Choice A (4).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what constitutes "distinct" intercepts.

Students might correctly identify that \(\mathrm{(x - d)}\) appears twice but incorrectly conclude that having any repeated factors somehow reduces the total count, or they might confuse the concept of distinct intercepts with the concept of simple vs. multiple roots.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

Success on this problem requires understanding that repeated factors in a polynomial's factored form indicate multiplicity at a root, not additional intercepts. The key insight is recognizing that "distinct x-intercepts" refers to different x-coordinate values where the graph crosses the axis, regardless of how many times each factor appears.