Question

How do I use long division to simplify $\dfrac{3{{x}^{3}}+4x+11}{{{x}^{2}}-3x+2}$ ?

Answer 1

Hint: To solve the above given equation $\dfrac{3{{x}^{3}}+4x+11}{{{x}^{2}}-3x+2}$ we will use long division method. In algebra polynomial long division is an algorithm for dividing polynomials of the same or lower degree it is known as long division method. The long division method looks like $\dfrac{f\left( x \right)}{g\left( x \right)}$ where $f\left( x \right)$is numerator and also known as dividend, and $g\left( x \right)$ is denominator and also known as divisor.

Complete step-by-step solution:
There are few steps which we follow while using long division method of polynomial:
1. Divide the first term of the numerator by the first term of the denominator, and put that in quotient.
2. Multiply the denominator by that answer and put that below the numerator.
3. subtract the both to get a new polynomial
4. repeat, the steps for the new polynomial.
Now for the given equation is
$\Rightarrow \dfrac{3{{x}^{3}}+4x+11}{{{x}^{2}}-3x+2}$
we have to multiply ${{x}^{2}}$ with $3x$to get $3{{x}^{3}}$, similarly we will repeat the process, we get
\[\begin{align}
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3x+9 \\
& \Rightarrow {{x}^{2}}-3x+2\left| \!{\overline {\,
\begin{align}
& 3{{x}^{3}}+4x+11 \\
& \underline{-3{{x}^{3}}+9{{x}^{2}}-6x} \\
& 9{{x}^{2}}-2x+11 \\
& \underline{-9{{x}^{2}}+2x-11} \\
& 25x-7 \\
\end{align} \,}} \right. \\
\end{align}\]
Here we get remainder is $25x-7$
Hence, we get a solution of a given equation by a long division method.

Note: We can also check if our answer is correct or not. We can follow these steps:
First do the division problem, then multiply the quotient with the divisor, and if there is a remainder, add it to the multiplication product, and then compare this answer to the dividend. They should be the same. For example, in the above problem,
$\begin{align}
  & \Rightarrow \left( {{x}^{2}}-3x+2 \right)\left( 3x+9 \right)+\left( 25x-7 \right) \\
 & \Rightarrow 3{{x}^{3}}+6x+9{{x}^{2}}-27x+18-9{{x}^{2}}+25x-7 \\
 & \Rightarrow 3{{x}^{3}}+4x+11 \\
\end{align}$
Which is equivalent to the dividend given in the question, hence our solution is correct. We can go wrong in the division part. Long division method is a simple method to solve the polynomials of those which are in ratio forms. And always remember to check your answer by using the above method. We always prefer to use a long division method because it is simple and no difficult and lengthy process.