 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-a/1/solution

The factor theorem gives a factor

x - 1

2 x^{3} + x^{2} - 5 x + 2 = (x - 1) (2 x^{2} + 3 x - 2) = (x - 1) (2 x - 1) (x + 2) .

Consequently all three roots of a polynomial are

x = 1, \frac{1}{2}, - 2.

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