Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 7 / a / ii

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- Solution ii

Solution

Let

P = 3 x^{4} - 5 x^{3} + 2 x^{2} - 3 x + 2

and

Q = 3 x^{3} + x^{2} + x - 2

. Applying polynomial long division gives the euclidean algorithm

P Q 3 x^{2} + x - 2 = (x - 2) Q + (3 x^{2} + x - 2), = x (3 x^{2} + x - 2) + (3 x - 2), = (x + 1) (3 x - 2) .

Thus their polynomial greatest common divisor is

3 x - 2

, up to a nonzero constant factor. Their unique common root of a polynomial is therefore

α = \frac{2}{3} .

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