Consider the expression -{63x^5 + 105x^4}.This expression is rewritten in factored form as \(\mathrm{r} \cdot \mathrm{x}^4(-9\mathrm{x} + 15)\), where...

1. TRANSLATE the problem information

• Given information:
  • Original expression: \(-63\mathrm{x}^5 + 105\mathrm{x}^4\)
  • Factored form: \(\mathrm{r} \cdot \mathrm{x}^4(-9\mathrm{x} + 15)\), where r is unknown
  • These two expressions are equal

• This tells us: \(-63\mathrm{x}^5 + 105\mathrm{x}^4 = \mathrm{r} \cdot \mathrm{x}^4(-9\mathrm{x} + 15)\)

2. SIMPLIFY to expand the factored form

• Distribute \(\mathrm{r} \cdot \mathrm{x}^4\) through the parentheses:
  \(\mathrm{r} \cdot \mathrm{x}^4(-9\mathrm{x} + 15) = \mathrm{r} \cdot \mathrm{x}^4 \cdot (-9\mathrm{x}) + \mathrm{r} \cdot \mathrm{x}^4 \cdot (15)\)
  \(= -9\mathrm{r}\mathrm{x}^5 + 15\mathrm{r}\mathrm{x}^4\)

• Now we have: \(-63\mathrm{x}^5 + 105\mathrm{x}^4 = -9\mathrm{r}\mathrm{x}^5 + 15\mathrm{r}\mathrm{x}^4\)

3. INFER the coefficient matching strategy

• Since both sides are polynomials and they're equal, coefficients of like terms must match
• For \(\mathrm{x}^5\) terms: \(-63 = -9\mathrm{r}\)
• For \(\mathrm{x}^4\) terms: \(105 = 15\mathrm{r}\)

4. SIMPLIFY to solve for r

• From the \(\mathrm{x}^5\) coefficient equation: \(-9\mathrm{r} = -63\)
• Divide both sides by -9:
  \(\mathrm{r} = -63/(-9)\)
  \(= 63/9\)
  \(= 7\)

5. Check your answer using the \(\mathrm{x}^4\) coefficients

• If \(\mathrm{r} = 7\), then \(15\mathrm{r} = 15(7) = 105\) ✓
• This matches the \(\mathrm{x}^4\) coefficient, confirming our answer

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not properly convert the word problem into the equation \(-63\mathrm{x}^5 + 105\mathrm{x}^4 = \mathrm{r} \cdot \mathrm{x}^4(-9\mathrm{x} + 15)\). Instead, they might try to factor the original expression directly or misunderstand what the problem is asking them to find.

This leads to confusion and abandoning the systematic approach, resulting in guessing.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly expand to get \(-9\mathrm{r}\mathrm{x}^5 + 15\mathrm{r}\mathrm{x}^4\) but make arithmetic errors when solving \(-9\mathrm{r} = -63\), such as forgetting about the negative signs or making division errors.

This may lead them to get an incorrect value like \(\mathrm{r} = -7\) or \(\mathrm{r} = 9\).

The Bottom Line:

This problem tests whether students can work backwards from a given factored form to find an unknown constant. The key insight is recognizing that the problem gives you the factored structure and asks you to determine what constant makes it work - it's not asking you to factor from scratch.