The expression 18x^3 + 12x^2 is equivalent to \(\mathrm{t}\mathrm{x}^2(3\mathrm{x} + 2)\), where t is a constant. What is the value...

1. INFER the solution strategy

• Given: \(\mathrm{18x^3 + 12x^2}\) is equivalent to \(\mathrm{tx^2(3x + 2)}\)
• Strategy choice: We can either expand the factored form and compare coefficients, or factor the original expression to match the given form

2. SIMPLIFY using expansion method

• Expand \(\mathrm{tx^2(3x + 2)}\):

\(\mathrm{tx^2(3x + 2) = tx^2(3x) + tx^2(2)}\)

\(\mathrm{= 3tx^3 + 2tx^2}\)

• Set expressions equal: \(\mathrm{18x^3 + 12x^2 = 3tx^3 + 2tx^2}\)

3. INFER coefficient relationships

• For equivalent polynomials, corresponding coefficients must be equal:

• Coefficient of \(\mathrm{x^3}\): \(\mathrm{18 = 3t}\)
• Coefficient of \(\mathrm{x^2}\): \(\mathrm{12 = 2t}\)

4. SIMPLIFY to find t

• From \(\mathrm{x^3}\) terms: \(\mathrm{18 = 3t}\) → \(\mathrm{t = 6}\)
• From \(\mathrm{x^2}\) terms: \(\mathrm{12 = 2t}\) → \(\mathrm{t = 6}\)
• Both equations confirm \(\mathrm{t = 6}\)

Answer: C) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make distribution errors when expanding \(\mathrm{tx^2(3x + 2)}\), particularly confusing the order of operations or making arithmetic mistakes.

For example, they might incorrectly expand as \(\mathrm{tx^2(3x) + tx^2(2) = 3tx^2 + 2tx}\) instead of \(\mathrm{3tx^3 + 2tx^2}\). This leads to comparing \(\mathrm{18x^3 + 12x^2 = 3tx^2 + 2tx}\), which creates impossible coefficient equations and causes confusion, leading to guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they can factor the original expression as an alternative approach, so they only try the expansion method. If they make errors during expansion, they have no way to check their work or try a different approach.

Some students might attempt to factor \(\mathrm{18x^3 + 12x^2}\) but pull out the wrong common factor (like \(\mathrm{2x^2}\) instead of \(\mathrm{6x^2}\)), leading to \(\mathrm{18x^3 + 12x^2 = 2x^2(9x + 6)}\) instead of \(\mathrm{6x^2(3x + 2)}\). This mismatch with the target form \(\mathrm{tx^2(3x + 2)}\) creates confusion about what t should be.

The Bottom Line:

This problem tests whether students can work flexibly with equivalent polynomial expressions using either expansion or factoring approaches, but success depends heavily on careful algebraic manipulation.