Find the remainder by using remainder theorem when polynomial $x^{3} - 3 x^{2} + x + 1$ is divided by $x - 1$ .

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Solution

The correct option is A $0$
$\frac{x^{3} - {3 x}^{2} + x + 1}{(x - 1)}$
$= \frac{x^{3} - x^{2} - 2 x^{2} + x + 1}{(x - 1)}$
$= \frac{x^{2} (x - 1) - 2 x^{2} + x + 1}{(x - 1)}$
$= x^{2} - \frac{2 x^{2} - x - 1}{(x - 1)}$
$= x^{2} - \frac{2 x^{2} - 2 x + 2 x - x - 1}{(x - 1)}$
$= x^{2} - \frac{2 x (x - 1) + x - 1}{(x - 1)}$
$= x^{2} - 2 x + 1$

$x^{3} - {3 x}^{2} + x + 1 = (x - 1) (x^{2} - 2 x - 1)$
$∴$ Remainder is $^{'} 0^{'}$

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