Find the polynomial which is to be added to $P (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ , so that the resulting polynomial is exactly divisible by $x^{2} + 2 x - 3$

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Solution

$x^{2} + 2 x - 3) x^{2} + 1 ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{4} + 2 x^{3} - 2 x^{2} + x - 1$
$x^{4} (-) + (-) 2 x^{3} - (+) 3 x^{2}$
$x^{2} + x - 1$ $
$x^{2} (-) + + (-) 2 x - (+) 3$
$- x + 2$
Thus, $r (x) = - x + 2 \Rightarrow {- r (x)} = x - 2$
Hence, we should add $(x - 2)$ to $P (x)$ , so that the resulting polynomial is exactly divisible by $g (x)$ .

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Find the polynomial which is to be added to P(x)=x4+2x3−2x2+x−1, so that the resulting polynomial is exactly divisible by x2+2x−3

Find the polynomial which is to be added to $P (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ , so that the resulting polynomial is exactly divisible by $x^{2} + 2 x - 3$