Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 18J / b

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- Solution b

Solution

Solve the quadratic resolvent:

u^{3}, v^{3} = \frac{- 27 b \pm - 27 Disc ( f )}{2} .

(1)

Choose cube roots with

uv = - 3 a

. Inverting the discrete Fourier transform gives

α_{1} = \frac{u + v}{3}, α_{2} = \frac{ω ^{2} u + ω v}{3}, α_{3} = \frac{ω u + ω ^{2} v}{3} .

(2)

These expressions use only square and cube roots. A general cubic is first translated to remove its quadratic term, proving the existence of a formula by radicals.

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18J
Paper 1
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2025
Past exam of the mathematics course of the University of Cambridge
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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 18J / b

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