\(\mathrm{h(x) = \frac{x^2 - 4x + 1}{x - 2}}\). For the function h defined above, what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = \frac{x^2 - 4x + 1}{x - 2}}\)
  • Need to find: \(\mathrm{h(-1)}\)

• What this tells us: We need to substitute \(\mathrm{x = -1}\) into the function and calculate the result.

2. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = -1}\) into \(\mathrm{h(x) = \frac{x^2 - 4x + 1}{x - 2}}\):

\(\mathrm{h(-1) = \frac{(-1)^2 - 4(-1) + 1}{(-1) - 2}}\)

• SIMPLIFY the numerator step by step:
  • \(\mathrm{(-1)^2 = 1}\) (careful: negative squared gives positive)
  • \(\mathrm{-4(-1) = +4}\) (negative times negative gives positive)
  • So: \(\mathrm{(-1)^2 - 4(-1) + 1 = 1 + 4 + 1 = 6}\)

• SIMPLIFY the denominator:
  • \(\mathrm{(-1) - 2 = -3}\)

3. SIMPLIFY the final division

• \(\mathrm{h(-1) = \frac{6}{-3} = -2}\)

Answer: B. -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when evaluating \(\mathrm{(-1)^2}\)

Many students incorrectly calculate \(\mathrm{(-1)^2 = -1}\) instead of \(\mathrm{(-1)^2 = +1}\). This leads to:

• Numerator becomes: \(\mathrm{-1 + 4 + 1 = 4}\) instead of 6
• Final answer becomes: \(\mathrm{\frac{4}{-3} = -\frac{4}{3}}\)

Since \(\mathrm{-\frac{4}{3} \approx -1.33}\) isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY reasoning: Students make arithmetic errors in the multiplication \(\mathrm{-4(-1)}\)

Some students incorrectly calculate \(\mathrm{-4(-1) = -4}\) instead of \(\mathrm{+4}\). This gives:

• Numerator: \(\mathrm{1 + (-4) + 1 = -2}\)
• Final answer: \(\mathrm{\frac{-2}{-3} = \frac{2}{3}}\)

Since \(\mathrm{\frac{2}{3}}\) isn't exactly matching any choice, they might select Choice C (-1/3) as the closest fraction.

The Bottom Line:

This problem tests careful execution of negative number operations more than complex mathematical reasoning. Students who rush through the arithmetic or lack confidence with negative number rules will struggle despite understanding the function evaluation concept.