The expression 63x^4 - 84x^3 + 21x^2 is equivalent to \(\mathrm{r}\mathrm{x}^2(9\mathrm{x}^2 - 12\mathrm{x} + 3)\), where r is a constant....

1. TRANSLATE the problem information

• Given: \(63\mathrm{x}^4 - 84\mathrm{x}^3 + 21\mathrm{x}^2\) is equivalent to \(\mathrm{r}\mathrm{x}^2 (9\mathrm{x}^2 - 12\mathrm{x} + 3)\)
• Find: The value of constant \(\mathrm{r}\)

2. SIMPLIFY the original expression by factoring

• Look for the greatest common factor in \(63\mathrm{x}^4 - 84\mathrm{x}^3 + 21\mathrm{x}^2\)
• All terms contain \(\mathrm{x}^2\), so factor it out:
  \(63\mathrm{x}^4 - 84\mathrm{x}^3 + 21\mathrm{x}^2 = \mathrm{x}^2(63\mathrm{x}^2 - 84\mathrm{x} + 21)\)

3. INFER the key relationship for equivalence

• Since \(\mathrm{x}^2(63\mathrm{x}^2 - 84\mathrm{x} + 21) = \mathrm{r}\mathrm{x}^2 (9\mathrm{x}^2 - 12\mathrm{x} + 3)\)
• The \(\mathrm{x}^2\) factors on both sides cancel out
• This means: \(63\mathrm{x}^2 - 84\mathrm{x} + 21 = \mathrm{r}(9\mathrm{x}^2 - 12\mathrm{x} + 3)\)

4. SIMPLIFY by comparing coefficients

• For equivalent quadratic expressions, corresponding coefficients must be proportional
• Coefficient of \(\mathrm{x}^2\): \(63 = 9\mathrm{r}\) → \(\mathrm{r} = 63/9 = 7\)
• Coefficient of \(\mathrm{x}\): \(-84 = -12\mathrm{r}\) → \(\mathrm{r} = -84/(-12) = 7\)
• Constant term: \(21 = 3\mathrm{r}\) → \(\mathrm{r} = 21/3 = 7\)

5. APPLY CONSTRAINTS to verify consistency

• All three ratios must give the same value of \(\mathrm{r}\)
• Since \(63/9 = 7\), \(-84/(-12) = 7\), and \(21/3 = 7\), our answer is consistent

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students may struggle with factoring the original expression correctly, either missing the \(\mathrm{x}^2\) common factor or making arithmetic errors when factoring out coefficients.

For example, they might factor as \(\mathrm{x}^2(63\mathrm{x}^2 - 84\mathrm{x}^2 + 21)\) (keeping \(\mathrm{x}^2\) in the second term) or make errors like \(\mathrm{x}(63\mathrm{x}^3 - 84\mathrm{x}^2 + 21\mathrm{x})\). Without the correct factored form, they cannot set up the proper equivalence relationship and will likely guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly factor out \(\mathrm{x}^2\) but make computational errors when calculating the ratios of coefficients.

They might compute \(63/9 = 6\) instead of 7, or confuse the signs when calculating \(-84/(-12)\). These arithmetic mistakes lead them to conclude that the coefficients aren't proportional or to select an incorrect value for \(\mathrm{r}\).

The Bottom Line:

This problem tests whether students can systematically factor polynomials and use the concept of equivalent expressions. Success depends on careful factoring followed by precise coefficient comparison.