Divide $x^{3} - 3 x^{2} + 3 x - 5$ by $x - 1 - x^{2}$ , and verify the division algorithm

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Solution

Lets do division

Verification of Division algorithm rule:
From Division:
Dividend $= x^{3} - 3 x^{2} + 3 x - 5$
divisor $= - x^{2} + x - 1$
Remainder $= - 3$
By Division algorithm,
Dividend $=$ divisor $\times$ Quotient $+$ Remainder
divisor $\times$ Quotient $+$ Remainder
$= (- x^{2} + x - 1) \times (- x + 2) + (- 3)$
$= - x^{2} (- x + 2) + x (- x + 2) - 1 (- x + 2) - 3$
$= x^{3} - 2 x^{2} - x^{2} + 2 x + x - 2 - 3$
$= x^{3} - 3 x^{2} + 3 x - 5$
$=$ Dividend
Therefore, Dividend $=$ divisor $\times$ Quotient $+$ Remainder
Hence, verified

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