 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/18i/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

f (T) = T^{3} + a T + b

f^{'} (T) = T^{2} + a .

A polynomial is separable exactly when it and its derivative have no common root. If

r

is a root of

f^{'}

, then

f (r) = r^{3} + a r + b = r (r^{2} + a) + b = b .

Thus $f$ and $f^{'}$ have a common root exactly when $b = 0$ . Therefore $f$ is a separable polynomial if and only if $b \neq = 0$ .

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