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Factorise:

x3 − 2x2 − x + 2

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Let p(x) = x3 − 2x2 − x + 2

All the factors of 2 have to be considered. These are ± 1, ± 2.

By trial method,

p(2) = (2)3 − 2(2)2 − 2 + 2

= 8 − 8 − 2 + 2 = 0

Therefore, (x − 2) is factor of polynomial p(x).

Let us find the quotient on dividing x3 − 2x2 − x + 2 by x − 2.

By long division,

It is known that,

Dividend = Divisor × Quotient + Remainder

∴ x3 − 2x2 − x + 2 = (x + 1) (x2 − 3x + 2) + 0 = (x + 1) [x2 − 2x − x + 2]

= (x + 1) [x (x − 2) − 1 (x − 2)] = (x + 1) (x − 1) (x − 2)

= (x − 2) (x − 1) (x + 1)

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