If the polynomial $3 x^{2} - x^{3} - 3 x + 5$ is divided by another polynomial $x - 1 - x^{2}$ , the remainder comes out to be $3$ , then quotient polynomial is

2 - x

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2 x - 1

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3 x + 4

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x - 2

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Solution

The correct option is D $x - 2$
According to division algorithm
Dividend $=$ Divisor $\times$ Quotient $+$ Remainder
$p (x) = g (x) \times q (x) + r (x)$
Putting the value in formula we get ,
${- x}^{3} + 3 x^{2} - 3 x + 5 = (x - 1 - x^{2}) \times q (x) + 3$
$\Rightarrow {- x}^{3} + 3 x^{2} - 3 x + 5 - 3 = (x - 1 - x^{2}) \times q (x)$
$\Rightarrow {- x}^{3} + 3 x^{2} - 3 x + 2 = (x - 1 - x^{2}) \times q (x)$
$\Rightarrow ({- x}^{3} + 3 x^{2} - 3 x + 2) = q (x) \times (x - 1 - x^{2})$
Dividing both sides by $(x - 1 - x^{2})$ , we get,
$\frac{({- x}^{3} + 3 x^{2} - 3 x + 2)}{(x - 1 - x^{2})} = q (x)$
So, $q (x) = x - 2$

Hence, option D.

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