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 Incoming links: Polynomial remainder theorem

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 3 / Solution 2026-10-18

By the polynomial remainder theorem, the first polynomial remainder is

f (1) = 23 - 2 p

. Equating it to

9

gives $p = 7$ .

For the separate polynomial long division, successively cancel the leading powers of the dividend. Subtracting

x^{2} (x - 1)

2 x^{2} + 2 x + 1

; subtracting

2 x (x - 1)

4 x + 1

; subtracting

4 (x - 1)

5

. Thus

x^{3} + x^{2} + 2 x + 1 = (x - 1) (x^{2} + 2 x + 4) + 5, quotient x^{2} + 2 x + 4, remainder 5.

The polynomial remainder also equals the dividend evaluated at

x = 1

.

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