In polynomial algebra, consider the cubic polynomial p·x^3 − 2x^2 − q·x + 18. If this polynomial is completely divisible by x^2 − 9 (that is, it has no remainder), find the ratio p : q.

Introduction / Context:
This problem again uses polynomial divisibility, but the goal is to find a ratio between two coefficients rather than their exact individual values. The factor x^2 − 9 has simple roots, and knowing that the cubic is divisible by this quadratic means each of those roots must also be roots of the cubic. This gives us equations relating p and q, from which we can determine their ratio p : q.

Given Data / Assumptions:
- The cubic polynomial is p·x^3 − 2x^2 − q·x + 18.
- It is divisible by x^2 − 9, which factors as (x − 3)(x + 3).
- Therefore 3 and −3 are roots of the cubic polynomial.
- We must determine the ratio p : q, not necessarily the individual values.

Concept / Approach:
Let f(x) = p·x^3 − 2x^2 − q·x + 18. Because f(x) is divisible by x^2 − 9, f(3) = 0 and f(−3) = 0. Substituting x = 3 and x = −3 into f(x) gives two equations involving p and q. Solving this system reveals a relationship between p and q. Even if the system does not pin down unique numeric values, it will give a proportional relationship, which is exactly what is asked for in the ratio p : q.

Step-by-Step Solution:
1) Define f(x) = p·x^3 − 2x^2 − q·x + 18. 2) Since x^2 − 9 = (x − 3)(x + 3), the roots are x = 3 and x = −3, so f(3) = 0 and f(−3) = 0. 3) Compute f(3): f(3) = p·27 − 2·9 − q·3 + 18 = 27p − 18 − 3q + 18 = 27p − 3q. 4) Set f(3) = 0, giving 27p − 3q = 0, or dividing by 3, 9p − q = 0, so q = 9p. 5) Compute f(−3): f(−3) = p·(−27) − 2·9 − q·(−3) + 18 = −27p − 18 + 3q + 18 = −27p + 3q. 6) Set f(−3) = 0, giving −27p + 3q = 0, or dividing by 3, −9p + q = 0, which is the same relation q = 9p. 7) Thus the ratio p : q satisfies q = 9p, so p : q = 1 : 9.

Verification / Alternative check:
We can choose a convenient value for p and compute q accordingly. For example, let p = 1, then q = 9. The polynomial becomes x^3 − 2x^2 − 9x + 18. Factor x^2 − 9 as (x − 3)(x + 3) and check if this divides the cubic. By performing polynomial division or grouping, you can confirm that x^3 − 2x^2 − 9x + 18 factors as (x^2 − 9)(x − 2), confirming that p : q = 1 : 9 is consistent with the divisibility condition.

Why Other Options Are Wrong:
Options B (1 : 3), C (3 : 1), D (9 : 1), and E (2 : 9) all represent different proportional relationships that do not satisfy the equations obtained from f(3) = 0 and f(−3) = 0. If any of these ratios were used, either f(3) or f(−3) would be non zero, meaning the polynomial would not be divisible by x^2 − 9. Only Option A reflects the correct ratio p : q = 1 : 9 that satisfies both conditions.

Common Pitfalls:
A common error is to compute f(3) correctly but forget to use f(−3), or vice versa, which can lead to incomplete or incorrect relationships. Another mistake is mishandling signs when substituting negative values like −3 into the polynomial. Dividing both sides of the equations by common factors also helps simplify the system and makes the ratio easier to identify. Carefully writing substitutions and checking arithmetic can prevent these errors.

Final Answer:
From the divisibility conditions at x = 3 and x = −3, we find that q = 9p, so the required ratio is 1:9.