 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iv/7/a/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

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) .

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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