 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/8/solution

(x - 1) (x - 2)

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

Comparing the

x^{2}

a = 1

. Substitution of

x = 1

gives

2 = - b

, while

x = 2

gives

5 = c

. Therefore

a = 1, b = - 2, c = 5.

The original identity holds for

x \neq = 1, 2

; evaluating the multiplied polynomial identity at those values is valid.

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