Common linear factor condition — repaired for single-correct option For polynomials f(x) = x^3 + c x^2 - x + 2c and g(x) = x^2 + c x - 2 over the rationals, which one of the following values of c makes their HCF (gcd) a linear polynomial (degree 1)?

Introduction / Context:
We want the greatest common divisor (over the rationals) of two polynomials to have degree 1. That means the polynomials share a common linear factor (i.e., they have at least one common rational/real root), but do not share a quadratic factor. We will derive c-values by eliminating c using a common root r.

Given Data / Assumptions:

• f(x) = x^3 + c x^2 - x + 2c.
• g(x) = x^2 + c x - 2.
• Degree-1 gcd ⇒ f and g have a common linear factor (a common root r).

Concept / Approach:
If r is a common root, g(r) = 0 ⇒ c = (2 - r^2) / r (r ≠ 0). Substitute this c into f(r) = 0 and solve for r, then back-calculate c. This is an elimination technique often quicker than computing a full resultant in exams.

Step-by-Step Solution:
From g(r) = 0: c = (2 - r^2) / r.Compute f(r) = r^3 + c r^2 - r + 2c. Substitute c:f(r) = r^3 + ((2 - r^2)/r) * r^2 - r + 2 * (2 - r^2)/r.Simplify: r^3 + (2 - r^2) r - r + (4 - 2 r^2)/r = r + (4 - 2 r^2)/r.Multiply by r: r^2 + 4 - 2 r^2 = 0 ⇒ 4 - r^2 = 0 ⇒ r = ±2.For r = 2: c = (2 - 4)/2 = -1. For r = -2: c = (2 - 4)/(-2) = +1.

Verification / Alternative check:
For c = 1, g(x) = x^2 + x - 2 = (x + 2)(x - 1) and f(x) = x^3 + x^2 - x + 2 = (x + 2)(x^2 - 1) = (x + 2)(x - 1)(x + 1). The common factor is linear (x + 2). For c = -1, g(x) = x^2 - x - 2 = (x - 2)(x + 1) and f(x) = x^3 - x^2 - x - 2 = (x - 2)(x^2 + 1); again a linear common factor exists. The question asks for a single value; choosing +1 suffices.

Why Other Options Are Wrong:
c = +2 or +4 do not create a shared linear factor (quick factor checks fail). c = 0 does not yield a common linear factor either. The repaired single-correct framing accepts +1; -1 is also a valid value but not among the intended single-choice answers originally due to a duplicate option issue in the source.

Common Pitfalls:
Attempting long division repeatedly instead of eliminating c via a common root. Also, forgetting to verify that the gcd is degree 1, not higher.

Final Answer:
+1