Question

How do you divide $\left( 5{{x}^{4}}+14{{x}^{3}}+9x \right)\div \left( {{x}^{2}}+3x \right)$ using long division?

Answer 1

Hint: We have to solve the given expression by using the long division method. For this first we will write the all terms in decreasing power of the variable order including the missing terms. Then we start dividing the terms by following the step by step procedure.

Complete step-by-step solution:
We have been given an expression $\left( 5{{x}^{4}}+14{{x}^{3}}+9x \right)\div \left( {{x}^{2}}+3x \right)$.
We have to divide the expression by using long division.
For long division method we have to follow some steps as follows:
First we divide the tens column dividend by the divisor. Then multiply the divisor by the quotient in the tens place column. Then subtract the product from the dividend and bring down the dividend in the ones column and repeat the process till no term is left in the divisor column.
Now, let us start dividing the terms by long division method. Then we will get
\[{{x}^{2}}+3x\overset{5{{x}^{2}}-x+3}{\overline{\left){\begin{align}
  & 5{{x}^{4}}+14{{x}^{3}}+9x \\
 & \underline{5{{x}^{4}}+15{{x}^{3}}} \\
 & 0-{{x}^{3}} \\
 & -{{x}^{3}}-3{{x}^{2}} \\
 & \underline{\overline{\begin{align}
  & 3{{x}^{2}}+9x \\
 & 3{{x}^{2}}+9x \\
\end{align}}} \\
 & 0 \\
\end{align}}\right.}}\]
So on dividing the given expression $\left( 5{{x}^{4}}+14{{x}^{3}}+9x \right)\div \left( {{x}^{2}}+3x \right)$ by using long division method we get the quotient as \[5{{x}^{2}}-x+3\].

Note: In algebraic long division methods we need to follow the same steps as we follow in the arithmetic. We need to perform division until no term is left in the divisor. We can also check our answer by using the formula that $\text{Dividend=quotient}\times \text{divisor+remainder}\text{.}$. So by substituting the values we can verify the answer.