TMUA Practice Question — Worked Solution

P2·Algebra & Surds·7.0Mock 0 · P2 · Q11

polynomial divisionfactor theoremconjecture from small casesperiodicityx^3 + 1

For each positive integer

n

, let

P_{n} (x) = x^{2 n} + x^{n} + 1.

For example,

P_{2} (x) = x^{4} + x^{2} + 1 = (x^{2} + x + 1) (x^{2} - x + 1)

.For which positive integers

n

P_{n} (x)

exactly divisible by

x^{2} - x + 1

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