Divide : $3x^{2} - x^{3} - 3x + 5$ by $x - 1 - x^{2}$.
(N/A) To divide the polynomial $p(x) = -x^{3} + 3x^{2} - 3x + 5$ by $g(x) = -x^{2} + x - 1$:
Step $1$: Divide the first term of the dividend $(-x^{3})$ by the first term of the divisor $(-x^{2})$,which gives $x$.
Step $2$: Multiply $x$ by the divisor $(-x^{2} + x - 1)$ to get $-x^{3} + x^{2} - x$. Subtract this from the dividend: $(-x^{3} + 3x^{2} - 3x + 5) - (-x^{3} + x^{2} - x) = 2x^{2} - 2x + 5$.
Step $3$: Divide the first term of the new dividend $(2x^{2})$ by the first term of the divisor $(-x^{2})$,which gives $-2$.
Step $4$: Multiply $-2$ by the divisor $(-x^{2} + x - 1)$ to get $2x^{2} - 2x + 2$. Subtract this from the current dividend: $(2x^{2} - 2x + 5) - (2x^{2} - 2x + 2) = 3$.
Thus,the quotient is $x - 2$ and the remainder is $3$.
Step $1$: Divide the first term of the dividend $(-x^{3})$ by the first term of the divisor $(-x^{2})$,which gives $x$.
Step $2$: Multiply $x$ by the divisor $(-x^{2} + x - 1)$ to get $-x^{3} + x^{2} - x$. Subtract this from the dividend: $(-x^{3} + 3x^{2} - 3x + 5) - (-x^{3} + x^{2} - x) = 2x^{2} - 2x + 5$.
Step $3$: Divide the first term of the new dividend $(2x^{2})$ by the first term of the divisor $(-x^{2})$,which gives $-2$.
Step $4$: Multiply $-2$ by the divisor $(-x^{2} + x - 1)$ to get $2x^{2} - 2x + 2$. Subtract this from the current dividend: $(2x^{2} - 2x + 5) - (2x^{2} - 2x + 2) = 3$.
Thus,the quotient is $x - 2$ and the remainder is $3$.
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