4a^2 + 20ab + 25b^2 Which of the following is a factor of the polynomial above?...

1. INFER the polynomial structure

• Looking at \(\mathrm{4a^2 + 20ab + 25b^2}\), I need to check if this follows a special pattern
• The first and last terms are both perfect squares, which suggests this might be a perfect square trinomial

2. INFER the perfect square trinomial pattern

• Perfect square trinomials have the form \(\mathrm{A^2 + 2AB + B^2 = (A + B)^2}\)
• Let me check if our polynomial fits this pattern

3. SIMPLIFY by identifying the square root terms

• First term: \(\mathrm{4a^2 = (2a)^2}\) → \(\mathrm{A = 2a}\)
• Last term: \(\mathrm{25b^2 = (5b)^2}\) → \(\mathrm{B = 5b}\)
• Middle term should be: \(\mathrm{2AB = 2(2a)(5b) = 20ab}\) ✓

4. INFER the factored form

• Since the pattern matches perfectly: \(\mathrm{4a^2 + 20ab + 25b^2 = (2a + 5b)^2}\)
• Therefore, \(\mathrm{(2a + 5b)}\) is a factor of the polynomial

Answer: B. 2a + 5b

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the perfect square trinomial pattern and instead trying to factor by grouping or other methods that don't work efficiently with this polynomial.

Students might see the coefficients 4, 20, and 25 and try to find common factors or use trial-and-error factoring methods. They get stuck because these approaches are inefficient for perfect square trinomials, leading to confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY execution: Recognizing it might be a perfect square but making errors in checking the middle term.

Students might correctly identify that \(\mathrm{4a^2 = (2a)^2}\) and \(\mathrm{25b^2 = (5b)^2}\), but then incorrectly calculate the middle term as \(\mathrm{2(2a)(5b) = 10ab}\) instead of \(\mathrm{20ab}\). This leads them to conclude it's not a perfect square and attempt other approaches.

This may lead them to select Choice A (a + b) if they try to oversimplify the factorization.

The Bottom Line:

Perfect square trinomials have a very specific pattern that, once recognized, makes factoring straightforward. The key insight is checking whether the middle term equals exactly twice the product of the square roots of the first and last terms.