The function f gives the product of a number, x, and a number that is 91 more than x. Which...

1. TRANSLATE the problem information

• Given information:
  • Function f gives the product of two quantities
  • First quantity: a number x
  • Second quantity: a number that is 91 more than x
• What this tells us: We need to multiply \(\mathrm{x}\) by \(\mathrm{(x + 91)}\)

2. TRANSLATE each component into math

• First number: \(\mathrm{x}\)
• Second number: "91 more than x" = \(\mathrm{x + 91}\)
• Product: \(\mathrm{x \times (x + 91)}\)

3. SIMPLIFY by expanding the product

• \(\mathrm{f(x) = x(x + 91)}\)
• \(\mathrm{f(x) = x^2 + 91x}\)

Answer: C. \(\mathrm{f(x) = x^2 + 91x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "91 more than x" as \(\mathrm{91x}\) instead of \(\mathrm{(x + 91)}\)

When they see "91 more than x," they might think this means "91 times x" because they're rushing or confusing "more than" with multiplication. This confusion makes it impossible to set up the correct product, leading to guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{x(x + 91)}\) but make algebraic expansion errors

They might expand \(\mathrm{x(x + 91)}\) as just \(\mathrm{x^2 + 91}\) (forgetting the middle term), which would lead them to select Choice B (\(\mathrm{x^2 + 91}\)). Or they might somehow think they need to add an extra 91 at the end and select Choice D (\(\mathrm{x^2 + 91x + 91}\)).

The Bottom Line:

This problem tests your ability to carefully translate word relationships into algebraic expressions. The key insight is recognizing that "91 more than x" means addition \(\mathrm{(x + 91)}\), not multiplication, and then correctly expanding the resulting product.