\(\mathrm{g(x) = \frac{3}{8}x + \frac{7}{6}}\) \(\mathrm{h(x) = 6x - 5}\) The functions g and h are defined by the equations...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = \frac{3}{8}x + \frac{7}{6}}\)
  • \(\mathrm{h(x) = 6x - 5}\)
  • Need to find \(\mathrm{g(x) \cdot h(x)}\)
• What this tells us: We need to multiply two linear expressions together

2. INFER the approach

• To find \(\mathrm{g(x) \cdot h(x)}\), substitute the function expressions and multiply
• This becomes: \(\mathrm{(\frac{3}{8}x + \frac{7}{6})(6x - 5)}\)
• Use distributive property (FOIL method) to expand

3. SIMPLIFY the multiplication

Set up the distributive property:

\(\mathrm{(\frac{3}{8}x + \frac{7}{6})(6x - 5) = \frac{3}{8}x(6x - 5) + \frac{7}{6}(6x - 5)}\)

Multiply each term:

• First term: \(\mathrm{\frac{3}{8}x \times 6x = \frac{18}{8}x^2 = \frac{9}{4}x^2}\)
• Second term: \(\mathrm{\frac{3}{8}x \times (-5) = -\frac{15}{8}x}\)
• Third term: \(\mathrm{\frac{7}{6} \times 6x = 7x}\)
• Fourth term: \(\mathrm{\frac{7}{6} \times (-5) = -\frac{35}{6}}\)

4. SIMPLIFY by combining like terms

Current expression: \(\mathrm{\frac{9}{4}x^2 - \frac{15}{8}x + 7x - \frac{35}{6}}\)

Combine the x terms: \(\mathrm{-\frac{15}{8} + 7 = -\frac{15}{8} + \frac{56}{8} = \frac{41}{8}}\)

Final result: \(\mathrm{\frac{9}{4}x^2 + \frac{41}{8}x - \frac{35}{6}}\)

Note: Based on the answer choices provided, there may be a typo in the original problem. If \(\mathrm{g(x) = \frac{3}{5}x + \frac{7}{6}}\) instead, the answer would be \(\mathrm{\frac{18}{5}x^2 + 4x - \frac{35}{6}}\).

Answer: D (assuming the intended function was \(\mathrm{g(x) = \frac{3}{5}x + \frac{7}{6}}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make arithmetic errors when multiplying fractions or combining like terms. For example, they might incorrectly calculate \(\mathrm{\frac{3}{8} \times 6 = \frac{18}{8}}\) but then mistakenly write this as 18/5, or they might struggle with combining \(\mathrm{-\frac{15}{8} + 7}\) by not finding a common denominator properly. This leads to selecting an incorrect coefficient for either the \(\mathrm{x^2}\) term or the x term, causing them to choose Choice A, B, or C.

Second Most Common Error:

Incomplete SIMPLIFY process: Students may correctly set up the multiplication but fail to fully combine like terms, particularly struggling with the fraction arithmetic needed to add \(\mathrm{-\frac{15}{8} + 7}\). They might leave terms separate or make sign errors during combination. This leads to confusion and guessing among the given choices.

The Bottom Line:

This problem tests both procedural fluency with polynomial multiplication and computational accuracy with fractions - two areas where small errors compound quickly and lead to wrong answers.