The function f is defined by \(\mathrm{f(x) = x^3 + px^2 + qx + r}\), where p, q, and r...

1. TRANSLATE the arithmetic progression information

• Given information:
  • The zeros form an arithmetic progression
  • Middle zero = 4
  • Common difference = 3

• What this tells us: In an arithmetic progression, each term equals the middle term ± (common difference)

2. INFER the three zeros from the progression

• Since middle zero is 4 and common difference is 3:
  • First zero: \(4 - 3 = 1\)
  • Middle zero: 4
  • Third zero: \(4 + 3 = 7\)

• Our three zeros are: 1, 4, and 7

3. INFER how to use the zeros to find the polynomial

• Key insight: If we know all the zeros, we can write the polynomial in factored form
• Since 1, 4, and 7 are zeros, by the factor theorem:
  \(\mathrm{f(x) = (x - 1)(x - 4)(x - 7)}\)

4. SIMPLIFY by expanding the factored form

• First, expand \(\mathrm{(x - 1)(x - 4)}\):
  \(\mathrm{(x - 1)(x - 4) = x^2 - 4x - x + 4 = x^2 - 5x + 4}\)

• Then multiply by \(\mathrm{(x - 7)}\):
  \(\mathrm{(x^2 - 5x + 4)(x - 7) = x^2(x - 7) - 5x(x - 7) + 4(x - 7)}\)
  \(\mathrm{= x^3 - 7x^2 - 5x^2 + 35x + 4x - 28}\)
  \(\mathrm{= x^3 - 12x^2 + 39x - 28}\)

5. TRANSLATE back to find r

• We have: \(\mathrm{f(x) = x^3 - 12x^2 + 39x - 28}\)
• Comparing with \(\mathrm{f(x) = x^3 + px^2 + qx + r}\)
• Therefore: \(\mathrm{r = -28}\)

Answer: -28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret the arithmetic progression, thinking the common difference applies in the wrong direction or confusing which zero is the 'middle' one.

For example, they might think the zeros are 4, 7, 10 (adding 3 each time from the middle) or 1, 4, 6 (inconsistent differences). This leads to factoring the wrong polynomial entirely, resulting in a completely different value for r and likely causes confusion leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the zeros as 1, 4, and 7 but make algebraic errors during the triple multiplication, particularly losing track of negative signs or incorrectly combining like terms.

For instance, they might expand to get \(\mathrm{x^3 - 12x^2 + 39x + 28}\) (wrong sign on the constant term) or make other coefficient errors. This systematic algebraic error leads them away from the correct answer of -28.

The Bottom Line:

This problem tests whether students can connect the abstract concept of arithmetic progressions to concrete polynomial zeros, then execute the multi-step algebraic expansion accurately. Success requires both conceptual understanding and careful algebraic manipulation.