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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