\(\mathrm{h(x) = x^3 + ax^2 + bx + c}\) The function h is defined above, where a, b, and c...

1. INFER the relationship between zeros and factors

• Given information:
  • Polynomial: \(\mathrm{h(x) = x^3 + ax^2 + bx + c}\)
  • Zeros: \(\mathrm{-5, 6,\ and\ 7}\)

• Key insight: If a polynomial has zeros at specific values, we can write it in factored form using those zeros.

2. INFER the factored form strategy

• Since -5, 6, and 7 are zeros, the polynomial must equal zero when x equals any of these values
• This means: \(\mathrm{h(x) = (x - (-5))(x - 6)(x - 7) = (x + 5)(x - 6)(x - 7)}\)
• Watch the signs carefully: zero at -5 means factor is \(\mathrm{(x + 5)}\), not \(\mathrm{(x - 5)}\)

3. SIMPLIFY through polynomial expansion

• First, multiply two factors: \(\mathrm{(x + 5)(x - 6)}\)
  \(\mathrm{= x^2 - 6x + 5x - 30}\)
  \(\mathrm{= x^2 - x - 30}\)

• Next, multiply by the third factor: \(\mathrm{(x^2 - x - 30)(x - 7)}\)
  \(\mathrm{= x^3 - 7x^2 - x^2 + 7x - 30x + 210}\)
  \(\mathrm{= x^3 - 8x^2 - 23x + 210}\)

4. INFER the coefficient values

• Comparing \(\mathrm{h(x) = x^3 - 8x^2 - 23x + 210}\) with \(\mathrm{h(x) = x^3 + ax^2 + bx + c}\):
  • \(\mathrm{a = -8}\)
  • \(\mathrm{b = -23}\)
  • \(\mathrm{c = 210}\)

Answer: 210

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that zeros can be used to construct the factored form of the polynomial.

Students often try to substitute the zeros directly into the original equation \(\mathrm{h(x) = x^3 + ax^2 + bx + c}\), creating a system of three equations with three unknowns. While this approach can work, it's much more complex and time-consuming than using the factored form. This leads to getting bogged down in complicated algebra or abandoning the systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when writing factors or during polynomial expansion.

The most common specific error is writing \(\mathrm{(x - 5)}\) instead of \(\mathrm{(x + 5)}\) for the zero at -5, or making arithmetic mistakes during the expansion process. For example, incorrectly expanding \(\mathrm{(x + 5)(x - 6)(x - 7)}\) might yield \(\mathrm{x^3 - 8x^2 - 23x - 210}\) instead of \(\mathrm{x^3 - 8x^2 - 23x + 210}\). This leads to selecting \(\mathrm{c = -210}\) instead of the correct answer \(\mathrm{c = 210}\).

The Bottom Line:

This problem tests whether students can connect the concept of polynomial zeros to factored form, then execute careful algebraic expansion. The key insight is recognizing that knowing the zeros gives you a direct path to the factored form, which is much more efficient than setting up systems of equations.