Question:The polynomial \(\mathrm{P(x) = 3x^3 - 45x^2 + 162x}\) has a factor of the form x - c, where c...

1. TRANSLATE the problem information

• Given: \(\mathrm{P(x) = 3x^3 - 45x^2 + 162x}\) has factors of the form \(\mathrm{(x - c)}\) where \(\mathrm{c \gt 0}\)
• Find: The largest possible value of \(\mathrm{c}\)

2. INFER the approach

• To find all possible values of \(\mathrm{c}\), I need to completely factor the polynomial
• Any linear factor \(\mathrm{(x - c)}\) will reveal a possible value of \(\mathrm{c}\)
• Complete factorization will show all such factors

3. SIMPLIFY by factoring out the GCF

• All terms contain the variable \(\mathrm{x}\)
• The coefficients are 3, 45, and 162
• GCF of coefficients: 3 (since \(\mathrm{45 = 3 \times 15}\) and \(\mathrm{162 = 3 \times 54}\))
• Factor out \(\mathrm{3x}\): \(\mathrm{P(x) = 3x(x^2 - 15x + 54)}\)

4. SIMPLIFY the quadratic factor

• Need to factor \(\mathrm{x^2 - 15x + 54}\)
• Look for two numbers that multiply to 54 and add to -15
• Factor pairs of 54: \(\mathrm{1 \times 54, 2 \times 27, 3 \times 18, 6 \times 9}\)
• Check sums: \(\mathrm{6 + 9 = 15}\), so \(\mathrm{(-6) + (-9) = -15}\) ✓
• Therefore: \(\mathrm{x^2 - 15x + 54 = (x - 6)(x - 9)}\)

5. INFER the complete factorization

• \(\mathrm{P(x) = 3x(x - 6)(x - 9)}\)
• The linear factors of the form \(\mathrm{(x - c)}\) with \(\mathrm{c \gt 0}\) are: \(\mathrm{(x - 6)}\) and \(\mathrm{(x - 9)}\)
• This gives possible values: \(\mathrm{c = 6}\) or \(\mathrm{c = 9}\)

6. APPLY CONSTRAINTS to find the maximum

• Both \(\mathrm{c = 6}\) and \(\mathrm{c = 9}\) satisfy the constraint \(\mathrm{c \gt 0}\)
• The largest possible value is \(\mathrm{c = 9}\)

Answer: B) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that complete factorization is necessary to find ALL possible linear factors. They might attempt to use methods like the Rational Root Theorem or guess-and-check with the answer choices, missing the systematic approach of complete factorization.

This leads to incomplete analysis and potentially selecting a smaller value like Choice A (6) after finding only one factor, or getting stuck and guessing randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when factoring the quadratic \(\mathrm{x^2 - 15x + 54}\). They might incorrectly identify the factor pair (such as using \(\mathrm{2 \times 27}\) instead of \(\mathrm{6 \times 9}\)) or make sign errors, leading to incorrect factors.

This may lead them to select Choice C (15) or other incorrect values based on their faulty factorization.

The Bottom Line:

This problem requires systematic polynomial factorization skills combined with the insight that complete factorization reveals all possible linear factors. Students who jump to shortcuts or make calculation errors will miss the correct systematic approach.