A polynomial \(\mathrm{P(x)}\) has zeros at -{8}, 0, and 5, where the zero at -{8} has multiplicity 2. If \(\mathrm{P(2...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(x)}\) has zeros at -8, 0, and 5
  • The zero at -8 has multiplicity 2
  • \(\mathrm{P(2 - 3t) = 0}\)
• We need to find the sum of all possible values of t

2. INFER the key relationship

• When \(\mathrm{P(2 - 3t) = 0}\), this means the expression (2 - 3t) must equal one of the zeros of \(\mathrm{P(x)}\)
• Since \(\mathrm{P(x)}\) has zeros at -8, 0, and 5, we need: \(\mathrm{2 - 3t = -8}\), OR \(\mathrm{2 - 3t = 0}\), OR \(\mathrm{2 - 3t = 5}\)
• Note: Even though -8 has multiplicity 2, this doesn't create additional equations—it's still just \(\mathrm{2 - 3t = -8}\)

3. CONSIDER ALL CASES by solving each equation

Case 1: \(\mathrm{2 - 3t = -8}\)

• Subtract 2: \(\mathrm{-3t = -10}\)
• Divide by -3: \(\mathrm{t = \frac{10}{3}}\)

Case 2: \(\mathrm{2 - 3t = 0}\)

• Subtract 2: \(\mathrm{-3t = -2}\)
• Divide by -3: \(\mathrm{t = \frac{2}{3}}\)

Case 3: \(\mathrm{2 - 3t = 5}\)

• Subtract 2: \(\mathrm{-3t = 3}\)
• Divide by -3: \(\mathrm{t = -1}\)

4. SIMPLIFY to find the final answer

• Sum all possible values: \(\mathrm{\frac{10}{3} + \frac{2}{3} + (-1)}\)
• Combine fractions: \(\mathrm{\frac{12}{3} + (-1) = 4 + (-1) = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't realize that \(\mathrm{P(2-3t) = 0}\) means 2-3t must equal the zeros of \(\mathrm{P(x)}\). Instead, they might try to construct the polynomial \(\mathrm{P(x) = x(x-5)(x+8)^2}\) and substitute 2-3t, leading to a complicated equation like \(\mathrm{(2-3t)(2-3t-5)(2-3t+8)^2 = 0}\). This creates unnecessary algebraic complexity and increases the chance of calculation errors, leading to confusion and guessing.

Second Most Common Error:

Incomplete CONSIDER ALL CASES: Students find one or two values of t but miss checking all three zeros. For example, they might only solve \(\mathrm{2-3t = 0}\) to get \(\mathrm{t = \frac{2}{3}}\), or forget to check the zero at 5. This leads to partial sums like \(\mathrm{\frac{2}{3}}\) instead of the complete answer of 3.

The Bottom Line:

The key insight is recognizing that when a polynomial equals zero, its input must be a zero of that polynomial. Students who miss this connection get bogged down in unnecessary polynomial expansion rather than using the direct relationship between function composition and zeros.