On dividing $x^{3} - 3 x^{2} + x + 2$ by polynomial $g (x)$ , the quotient and remainder were $x - 2$ and $4 - 2 x$ respectively, then $g (x)$ is

x^{2} + x + 1

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x^{2} + x - 1

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x^{2} - x - 1

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x^{2} - x + 1

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Solution

The correct option is C $x^{2} - x + 1$
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 ,
$x^{3} - 3 x^{2} + x + 2 = g (x) \times (x - 2) + 4 - 2 x$
Adding $2 x$ and subtracting $4$ both sides we get,
$x^{3} - 3 x^{2} + x + 2 + 2 x - 4 = g (x) \times (x - 2)$
$\Rightarrow (x^{3} - 3 x^{2} + 3 x - 2) = g (x) \times (x - 2)$
Dividing $(x - 2)$ we get,
$\frac{x^{3} - 3 x^{2} + 3 x - 2}{x - 2} = g (x)$
So, $g (x) = x^{2} - x + 1.$

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On dividing x3−3x2+x+2 by polynomial g(x), the quotient and remainder were x−2 and 4−2x respectively, then g(x) is

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