Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 1

Table of contents
- Solution 1

Solution

Write the remaining quadratic polynomial as

x^{2} + b x + c

. Comparing coefficients in

(x - 2) (x^{2} + b x + c)

with the given cubic polynomial gives

b - 2 = - 1

and

- 2 c = - 2

, hence

b = c = 1

. The coefficient of

x

is then

c - 2 b = - 1

, as required. Thus

x^{3} - x^{2} - x - 2 = (x - 2) (x^{2} + x + 1), other factor x^{2} + x + 1.

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