 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/8/solution

(x + 1) (x + 2)^{2}

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

x = - 1

, this gives

a = 2

. Comparing the coefficients of

x^{2}

then gives

a + b = 1

, so

a = 2, b = - 1.

The remaining coefficients agree:

4 a + 3 b - 5 = 0

and

4 a + 2 b - 5 = 1

. The original rational identity is understood away from its poles

- 1

and

- 2

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