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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = px^2 + qx + r}\) (quadratic function)
  • Axis of symmetry: \(\mathrm{x = 4}\)
  • Passes through point \(\mathrm{(1, 0)}\)
  • p is an integer greater than 1
• Need to find: possible value of \(\mathrm{p + q}\)

2. INFER the key relationship using axis of symmetry

• For any quadratic \(\mathrm{ax^2 + bx + c}\), the axis of symmetry is \(\mathrm{x = -\frac{b}{2a}}\)
• In our case: \(\mathrm{x = -\frac{q}{2p} = 4}\)
• Solving: \(\mathrm{-q = 8p}\), therefore \(\mathrm{q = -8p}\)
• This is the crucial relationship that connects p and q!

3. INFER additional constraints from the point condition

• Since \(\mathrm{g(1) = 0}\), we have: \(\mathrm{p(1)^2 + q(1) + r = 0}\)
• This simplifies to: \(\mathrm{p + q + r = 0}\)
• Substituting \(\mathrm{q = -8p}\): \(\mathrm{p + (-8p) + r = 0}\)
• Therefore: \(\mathrm{r = 7p}\)

4. SIMPLIFY to find p + q

• Now we can express \(\mathrm{p + q}\) directly:
• \(\mathrm{p + q = p + (-8p) = -7p}\)

5. APPLY CONSTRAINTS to narrow the possibilities

• Since p is an integer greater than 1: \(\mathrm{p \geq 2}\)
• Therefore: \(\mathrm{p + q = -7p \leq -7(2) = -14}\)
• More importantly: \(\mathrm{p + q}\) must be a multiple of -7

6. APPLY CONSTRAINTS to check answer choices

• Looking for multiples of -7 among the choices:
  • (A) -33 ÷ (-7) = 4.71... (not an integer)
  • (B) -24 ÷ (-7) = 3.43... (not an integer)
  • (C) -21 ÷ (-7) = 3 ✓ (\(\mathrm{p = 3}\), which satisfies \(\mathrm{p \gt 1}\))
  • (D) -16 ÷ (-7) = 2.29... (not an integer)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not connect the axis of symmetry condition to the relationship between coefficients. They might try to use the vertex form or attempt to find r first, getting bogged down in unnecessary algebra. This leads to confusion and abandoning systematic solution, often resulting in guessing.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students might correctly find that \(\mathrm{p + q = -7p}\) but fail to recognize that this means \(\mathrm{p + q}\) must be a multiple of -7. They could calculate specific values for small integers (like \(\mathrm{p = 2}\) giving \(\mathrm{p + q = -14}\)) and then select the closest answer choice, potentially choosing (D) -16 instead of the correct answer.

The Bottom Line:

This problem tests whether students can translate geometric properties (axis of symmetry) into algebraic relationships between coefficients, then apply integer constraints systematically rather than just plugging in numbers.