Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 1

Table of contents
- a 1
  - Solution a
- b 1
  - Solution b

a

Solution

The sum-of-cubes polynomial factorization is

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) .

Multiplication cancels the two mixed cubic terms and leaves precisely the two cubes.

b

Solution

The polynomial factors as

x^{3} + x^{2} - 16 x + 20 = (x - 2)^{2} (x + 5) .

Thus the repeated polynomial root is $x = 2$ , of multiplicity two. Its repetition can also be checked from

p (2) = p^{'} (2) = 0

and

p^{''} (2) = 14 \neq = 0

.

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