The polynomial which when divided by $-x^{2}+x-1$
Question:

The polynomial which when divided by $-x^{2}+x-1$ gives a quotient $x-2$ and remainder 3 , is

(a) $x^{3}-3 x^{2}+3 x-5$

(b) $-x^{3}-3 x^{2}-3 x-5$

(c) $-x^{3}+3 x^{2}-3 x+5$

(d) $x^{3}-3 x^{2}-3 x+5$

Solution:

We know that

$f(x)=g(x) q(x)+r(x)$

$=\left(-x^{2}+x-1\right)(x-2)+3$

$=-x^{3}+x^{2}-x+2 x^{2}-2 x+2+3$

$=-x^{3}+x^{2}+2 x^{2}-x-2 x+2+3$

$=-x^{3}+3 x^{2}-3 x+5$

Therefore,

The polynomial which when divided by $-x^{2}+x-1$ gives a quotient $x-2$ and remainder 3 , is $-x^{3}+3 x^{2}-3 x+5$

Hence, the correct choice is $(c)$.

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