Question

How do you divide $\dfrac{6{{x}^{4}}+4{{x}^{3}}-12{{x}^{2}}+2x-7}{5x-2}$?

Answer 1

Hint: We will 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 answer:
We have been given an expression $\dfrac{6{{x}^{4}}+4{{x}^{3}}-12{{x}^{2}}+2x-7}{5x-2}$.
We have to divide the expression.
We will solve the given expression by using the long division method.
For long division method first we will arrange the terms in decreasing power of the variable order including the missing terms. Then we get the dividend as \[6{{x}^{4}}+4{{x}^{3}}-12{{x}^{2}}+2x-7\]. In algebraic long division method we need to follow same steps as we follow in the arithmetic division.
Now, let us start dividing the terms by long division method. Then we will get
\[5x-2\overset{\dfrac{6}{5}{{x}^{3}}+\dfrac{32}{25}{{x}^{2}}-\dfrac{236}{125}x-\dfrac{222}{625}-\dfrac{4819}{625\left( 5x-2 \right)}}{\overline{\left){\begin{align}
  & 6{{x}^{4}}+4{{x}^{3}}-12{{x}^{2}}+2x-7 \\
 & \underline{6{{x}^{4}}-\dfrac{12}{5}{{x}^{3}}} \\
 & \dfrac{32}{5}{{x}^{3}}-12{{x}^{2}} \\
 & \underline{\dfrac{32}{5}{{x}^{3}}-\dfrac{64}{25}{{x}^{2}}} \\
 & \dfrac{236}{25}{{x}^{2}}-2x \\
 & \underline{\dfrac{236}{25}{{x}^{2}}+\dfrac{472}{125}x} \\
 & \dfrac{222}{125}x-7 \\
 & \underline{\dfrac{222}{125}x+\dfrac{444}{625}} \\
 & \dfrac{4819}{625} \\
 & \underline{\dfrac{4819}{625}} \\
 & 0 \\
\end{align}}\right.}}\]
So on dividing the given expression by using long division method we get the quotient as \[\dfrac{6}{5}{{x}^{3}}+\dfrac{32}{25}{{x}^{2}}-\dfrac{236}{125}x-\dfrac{222}{625}-\dfrac{4819}{625\left( 5x-2 \right)}\] .

Note: 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. The point to be noted is that if the terms are not in decreasing power of variable then we need to arrange the terms in decreasing power of the variable order including the missing terms.