Answer
The value of $\left( {{x}^{3}}-4{{x}^{2}}-2x+5 \right)\div \left( x-1 \right)$is ${{x}^{2}}-3x-5$.
Work Step by Step
Consider the expression:
$\left( {{x}^{3}}-4{{x}^{2}}-2x+5 \right)\div \left( x-1 \right)$
The constant term of the divisor$-1$ is written to the left.
The coefficients of the dividend 1,$-4,-2$, 5 are written to the right.
Evaluate the value of $\left( {{x}^{3}}-4{{x}^{2}}-2x+5 \right)\div \left( x-1 \right)$ using synthetic division as follows.
$\begin{array}{c|rrrr}{1}&1&-4&-2&{5}\\&&1&-3&{1} \cdot {-5}={-5}\\\hline&1&-3&{-5}&\left({5}\right)+{-5}={0}\end{array}
$
The obtained numbers are the coefficients of a polynomial and the last digit is the remainder.
So, the polynomial has coefficients $1,-3,-5$is ${{x}^{2}}-3x-5$ and the remainder is 0.
Therefore, the value of$\left( {{x}^{3}}-4{{x}^{2}}-2x+5 \right)\div \left( x-1 \right)$ is ${{x}^{2}}-3x-5$.
