Question

On dividing ${{x}^{3}}-3{{x}^{2}}+x+2$ by a polynomial $g(x)$ the quotient and remainder were $x-2$ and $-2x+4$ respectively. Find $g(x)$.

Answer 1

Hint: Use the formula $\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder}$ to determine $g(x)$. First subtract the remainder from dividend then divide the obtained equation by quotient to get the divisor $g(x)$.

Complete step-by-step answer:

There is an algorithm called Euclid division lemma or Euclid division algorithm. This algorithm states that, if we have two positive integers $a$ and $b$, then there exist unique integers $q$ and $r$ which satisfies the condition $a=bq+r$ where $0\le r\le b$.
Now, let us come to the question.
Since, $\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder}$. Therefore,
${{x}^{3}}-3{{x}^{2}}+x+2$$=g(x)\times (x-2)+4-2x$
$\begin{align}
  & {{x}^{3}}-3{{x}^{2}}+x+2-4+2x=g(x)\times (x-2) \\
 & {{x}^{3}}-3{{x}^{2}}+3x-2=g(x)\times (x-2) \\
 & g(x)=\dfrac{{{x}^{3}}-3{{x}^{2}}+3x-2}{(x-2)} \\
\end{align}$
There is an easy method to simplify this type of division. Follow the steps given to divide the numerator with the given denominator.
Step 1: The numerator is a cubic polynomial and $x-2$ is its factor. So, write $x-2$ three times with some space between all of them.
$(x-2)\text{ }(x-2)\text{ }(x-2)$
Step 2: Get the first term of numerator by multiplying ${{x}^{2}}$ with $x-2$.
${{x}^{2}}(x-2)\text{ }(x-2)\text{ }(x-2)$
Step 3: Now balance the terms by mathematical operations needed.
$\begin{align}
  & {{x}^{2}}(x-2)\text{ }-x(x-2)\text{ +1}(x-2) \\
 & ={{x}^{2}}(x-2)-x(x-2)\text{+1}(x-2) \\
\end{align}$
Step 4: Take $x-2$ common and write the terms used for balancing in a single bracket.
$(x-2)\left( {{x}^{2}}-x+1 \right)$
Hence, $g(x)$$={{x}^{2}}-x+1$.

Note: We can also use a long division method to find $g(x)$ but that would consume a lot of space and every time we have to change the sign from ‘plus to minus’ or ‘minus to plus’. There is a lot of chance to get confused while using the long division method.