Which expression is equivalent to 121x^2 + 110x + 25?\((11\mathrm{x} + 5)(11\mathrm{x} + 5)\)\((11\mathrm{x} - 5)(11\mathrm{x} - 5)\)\((11\mathrm{x} +...

1. INFER the factoring approach

• Looking at \(121\mathrm{x}^2 + 110\mathrm{x} + 25\), notice that both the first and last terms are perfect squares
• Given information:
  • \(121\mathrm{x}^2 = (11\mathrm{x})^2\)
  • \(25 = 5^2\)
  • Middle term is \(110\mathrm{x}\)
• This suggests we should check for a perfect square trinomial pattern

2. INFER the perfect square trinomial pattern

• Perfect square trinomials have the form \(\mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\)
• If \(\mathrm{a} = 11\mathrm{x}\) and \(\mathrm{b} = 5\), then the middle term should be \(2\mathrm{ab} = 2(11\mathrm{x})(5) = 110\mathrm{x}\)
• Since this matches our middle term exactly, we have a perfect square trinomial

3. SIMPLIFY using the perfect square formula

• Apply the formula: \(\mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2 = (\mathrm{a} + \mathrm{b})^2\)
• Therefore: \(121\mathrm{x}^2 + 110\mathrm{x} + 25 = (11\mathrm{x} + 5)^2\)
• Since \((11\mathrm{x} + 5)^2 = (11\mathrm{x} + 5)(11\mathrm{x} + 5)\), our answer is \((11\mathrm{x} + 5)(11\mathrm{x} + 5)\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the perfect square trinomial pattern

Students often don't notice that \(121\mathrm{x}^2\) and \(25\) are perfect squares, or they don't connect this observation to the perfect square trinomial formula. Instead, they might try to factor by grouping or use trial-and-error with different factor pairs. This leads to confusion and time-consuming calculations that may result in giving up and guessing.

Second Most Common Error:

Conceptual confusion about signs: Incorrectly applying negative signs

Some students recognize the perfect square pattern but get confused about whether the signs should be positive or negative. They might think that since we often see differences like \(\mathrm{a}^2 - \mathrm{b}^2\), the factored form should involve negative signs. This may lead them to select Choice B: \((11\mathrm{x} - 5)(11\mathrm{x} - 5)\).

The Bottom Line:

The key insight is recognizing that when both the first and last terms of a trinomial are perfect squares, and the middle term equals twice the product of their square roots, you have a perfect square trinomial that factors as \((\mathrm{a} + \mathrm{b})^2\) or \((\mathrm{a} - \mathrm{b})^2\).