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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/5/a/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 5 / a / Solution by

Codex  0 2026-10-18

Writing the polynomial as

p (x)

, direct evaluation gives

p (0) = - 6

and

p (3) = 42

. A polynomial is continuous, so the intermediate value theorem guarantees a polynomial root between them. Trying the integer

2

gives

16 + 12 - 22 - 6 = 0

, hence $x = 2$ is a root.

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