 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/1/solution

The known root gives a factor

x - 1

2 x^{2} - 7 x + 6 = (2 x - 3) (x - 2)

, so

2 x^{3} - 9 x^{2} + 13 x - 6 = (x - 1) (2 x - 3) (x - 2) .

Consequently all the roots are

x = 1, x = \frac{3}{2}, x = 2.

The three linear factors account for the full degree of the polynomial, so there are no further complex roots.

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