Real factorization condition: For which values of k does the quadratic polynomial 3z^2 + 5z + k factor over the reals (i.e., has real linear factors)?

Introduction / Context:
A quadratic az^2 + bz + c factors into real linear factors iff its discriminant Δ = b^2 − 4ac is nonnegative. This is the necessary and sufficient condition for real roots (counting multiplicity). Apply it directly to 3z^2 + 5z + k.

Given Data / Assumptions:

• a = 3, b = 5, c = k.
• Real coefficients; want real roots.

Concept / Approach:
Impose b^2 − 4ac ≥ 0 ⇒ 25 − 4*3*k ≥ 0 ⇒ 25 − 12k ≥ 0 ⇒ k ≤ 25/12.

Step-by-Step Solution:

Verification / Alternative check:
At k = 25/12, Δ = 0, giving a repeated real root; for k less than this value, Δ > 0 gives two distinct real roots. For k > 25/12, Δ < 0, so no real factorization.

Why Other Options Are Wrong:

• k ≤ 25/6 or k ≥ 25/6: Wrong threshold; uses 4ac with a mistaken coefficient.
• k ≥ 25/12: Reverses the inequality; would imply nonreal roots for most values.

Common Pitfalls:
Arithmetic errors in computing 4ac or reversing the inequality sign when moving terms. Keep Δ ≥ 0 straight.

Final Answer: