Divide ${{x}^{3}}-6{{x}^{2}}+11x-6$ by $x-2$ and verify the division algorithm.

Hint: We start solving the problem by finding the dividend and divisor of the division process. We then perform the long-division process for the dividend with the divisor to obtain quotient and remainder. We then recall the division algorithm as dividend = divisor $\times $ quotient + remainder and substitute the divisor, quotient and remainder in this algorithm to check whether it is satisfying it or not.

Complete step by step answer:
According to the problem, we need to divide ${{x}^{3}}-6{{x}^{2}}+11x-6$ by $x-2$ and verify it with the division algorithm.
Here the dividend is ${{x}^{3}}-6{{x}^{2}}+11x-6$ and the divisor is $x-2$. We need to find the quotient and remainder of the division process between them.
Let us perform the division by long-division method as shown below.

So, we have found the quotient as ${{x}^{2}}-4x+3$ and the remainder as 0.
We know that the division algorithm is defined as dividend = divisor $\times $ quotient + remainder. Let us verify this with the obtained quotient and remainder.
So, we have dividend = $\left( \left( x-2 \right)\times \left( {{x}^{2}}-4x+3 \right) \right)+0$.
$\Rightarrow $ Dividend = $x\times \left( {{x}^{2}}-4x+3 \right)-2\times \left( {{x}^{2}}-4x+3 \right)$.
$\Rightarrow $ Dividend = ${{x}^{3}}-4{{x}^{2}}+3x-2{{x}^{2}}+8x-6$.
$\Rightarrow $ Dividend = ${{x}^{3}}-6{{x}^{2}}+11x-6$.
We can see that the divisor, dividend, quotient and remainder verified the division algorithm.

Note: We can also find the denominator of the division process by substituting $x=2$ in the dividend ${{x}^{3}}-6{{x}^{2}}+11x-6$. We can also factorize the given dividend with one factor as $x-2$ to find the quotient (if the remainder is zero) of the division process. We can also find the other factors of the dividend by equating dividend to zero. Similarly, we can expect problems to find the value we obtain on substituting $x=-1$ in the dividend.