Let \(\mathrm{P(t, s) = 9t^4 - 30t^2s + 25s^2}\). Which of the following is a factor of \(\mathrm{P(t, s)}\)? t^2...

1. TRANSLATE the problem information

• Given: \(\mathrm{P(t, s) = 9t^4 - 30t^2s + 25s^2}\)
• Find: Which expression from the choices is a factor of \(\mathrm{P(t, s)}\)

2. INFER the factoring approach

• Notice that this polynomial involves \(\mathrm{t^4}\) and \(\mathrm{t^2}\), suggesting we can treat it as quadratic in \(\mathrm{t^2}\)
• The structure \(\mathrm{9t^4 - 30t^2s + 25s^2}\) resembles a perfect square trinomial
• Strategy: Use substitution \(\mathrm{u = t^2}\) to simplify recognition

3. SIMPLIFY using substitution

• Let \(\mathrm{u = t^2}\), then: \(\mathrm{P = 9u^2 - 30us + 25s^2}\)
• Now we have a quadratic expression in u with parameter s

4. INFER the perfect square pattern

• Check if this fits \(\mathrm{a^2 - 2ab + b^2 = (a - b)^2}\)
• First term: \(\mathrm{9u^2 = (3u)^2}\)
• Last term: \(\mathrm{25s^2 = (5s)^2}\)
• Middle term: \(\mathrm{-30us = -2(3u)(5s)}\) ✓

5. SIMPLIFY the factorization

• \(\mathrm{P = (3u - 5s)^2}\)
• Substitute back \(\mathrm{u = t^2}\): \(\mathrm{P(t, s) = (3t^2 - 5s)^2}\)
• Therefore, \(\mathrm{3t^2 - 5s}\) is a factor

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the perfect square trinomial structure because they're intimidated by the two variables and the fourth power of t.

Many students see \(\mathrm{9t^4 - 30t^2s + 25s^2}\) and think it's too complex to factor systematically. They might try to guess and check with the answer choices instead of recognizing the underlying pattern. This leads to confusion and potentially guessing among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize it might be a perfect square but make sign errors in the factorization.

They might correctly identify that \(\mathrm{9t^4 = (3t^2)^2}\) and \(\mathrm{25s^2 = (5s)^2}\), but incorrectly handle the middle term's sign. This could lead them to think the factorization is \(\mathrm{(3t^2 + 5s)^2}\) and select Choice C \(\mathrm{(3t^2 + 5s)}\) instead of recognizing that the negative middle term requires \(\mathrm{(3t^2 - 5s)^2}\).

The Bottom Line:

This problem tests whether students can see past the complexity of two variables and recognize fundamental algebraic patterns. The key insight is treating this as a quadratic in \(\mathrm{t^2}\) rather than getting overwhelmed by the polynomial's appearance.