Question:16m^2 - 40mn + 25n^2Which of the following is a factor of the polynomial above?

1. INFER the factoring strategy

• Looking at \(16\mathrm{m}^2 - 40\mathrm{mn} + 25\mathrm{n}^2\), I notice this might be a perfect square trinomial
• Perfect square trinomials have the form \(\mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})^2\)
• Let me check if this polynomial fits that pattern

2. SIMPLIFY by identifying the components

• First term: \(16\mathrm{m}^2 = (4\mathrm{m})^2\) → so \(\mathrm{a} = 4\mathrm{m}\)
• Last term: \(25\mathrm{n}^2 = (5\mathrm{n})^2\) → so \(\mathrm{b} = 5\mathrm{n}\)
• Middle term should be: \(-2\mathrm{ab} = -2(4\mathrm{m})(5\mathrm{n}) = -40\mathrm{mn}\) ✓

3. SIMPLIFY by applying the perfect square formula

• Since all three terms match the pattern \(\mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2\):
• \(16\mathrm{m}^2 - 40\mathrm{mn} + 25\mathrm{n}^2 = (4\mathrm{m} - 5\mathrm{n})^2\)
• This means \((4\mathrm{m} - 5\mathrm{n})\) is a factor of the polynomial

Answer: B \((4\mathrm{m} - 5\mathrm{n})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize the perfect square trinomial pattern and instead attempt to factor by grouping or trial-and-error methods. They may try to find two binomials that multiply to give the original expression without realizing it's a special case. This leads to confusion and often causes them to abandon systematic solution and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize it might be a perfect square but make calculation errors when checking the middle term. For example, they might incorrectly calculate \(-2(4\mathrm{m})(5\mathrm{n})\) as \(-20\mathrm{mn}\) instead of \(-40\mathrm{mn}\), conclude it's not a perfect square, and then select an incorrect answer like Choice A \((2\mathrm{m} - 5\mathrm{n})\) thinking it might work through basic factoring attempts.

The Bottom Line:

Success on this problem hinges on pattern recognition - specifically identifying perfect square trinomials. Students who memorize and can apply the \(\mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})^2\) pattern will solve this efficiently, while those who don't recognize it will struggle with more complex factoring approaches.