Question

How do I use long division to simplify $\dfrac{2{{x}^{3}}+4{{x}^{2}}-5}{x+3}$ ?

Answer 1

Hint: To solve the above given equation $\dfrac{2{{x}^{3}}+4{{x}^{2}}-5}{x+3}$, we will long division method. In algebra, polynomial long division is an algorithm for dividing a polynomial 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. 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 answer.
2. Multiply the denominator by that answer and put that below the numerator.
3. Subtract to create a new polynomial.
4. Repeat, using the new polynomial.

Complete step by step solution:
 Here the given equation is,
$\Rightarrow \dfrac{2{{x}^{3}}+4{{x}^{2}}-5}{x+3}$
What we have to multiply $x$ by to get $2{{x}^{3}}$, we have to multiply $x$ with $2{{x}^{2}}$, similarly we will repeat the process, we get

\[x+3\overset{2{x}^{2}-2x+6}{\overline{\left){\begin{align}
& 2{{x}^{3}}+4{{x}^{2}}-5 \\
& \underline{2{{x}^{3}}+6{{x}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\
& \underline{\begin{align}
& \,\,\,\,\,\,\,\,-2{x}^{2}-5 \\
& \,\,\,\,\,\,\,\,-2{{x}^{2}}-6x\,\,\,\,\,\,\,\,\,\,\,
\end{align}} \\
& \underline{\begin{align}
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6x-5 \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6x+18\,\,\, \\
\end{align}} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-23 \\
\end{align}}\right.}}\]
 Here we get remainder $-23$
Hence we get a solution of a given equation by a long division method.

Note:
We can also check our answer, is it 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 to the answer to the dividend. They should be the same.
$\begin{align}
  & \Rightarrow \left( x+3 \right)\left( 2{{x}^{2}}-2x+6 \right)-23 \\
 & \Rightarrow 2{{x}^{3}}-2{{x}^{2}}+6x+6{{x}^{2}}-6x+18-23 \\
 & \Rightarrow 2{{x}^{3}}+4{{x}^{2}}-5 \\
 & \\
\end{align}$
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.