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Divide $$$- 2 a^{2} + 5 a - 8$$$ by $$$7 a^{2} - 3 a - 1$$$

The calculator will divide $$$- 2 a^{2} + 5 a - 8$$$ by $$$7 a^{2} - 3 a - 1$$$ using long division, with steps shown.

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Your Input

Find $$$\frac{- 2 a^{2} + 5 a - 8}{7 a^{2} - 3 a - 1}$$$ using long division.

Solution

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\7 a^{2}- 3 a-1&- 2 a^{2}+5 a-8\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{- 2 a^{2}}{7 a^{2}} = - \frac{2}{7}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- \frac{2 \left(7 a^{2}- 3 a-1\right)}{7} = - 2 a^{2}+\frac{6 a}{7}+\frac{2}{7}$$$.

Subtract the dividend from the obtained result: $$$\left(- 2 a^{2}+5 a-8\right) - \left(- 2 a^{2}+\frac{6 a}{7}+\frac{2}{7}\right) = \frac{29 a}{7}- \frac{58}{7}$$$.

$$\begin{array}{r|rrr:c}&{\color{DeepPink}- \frac{2}{7}}&&&\\\hline\\{\color{Magenta}7 a^{2}}- 3 a-1&{\color{DeepPink}- 2 a^{2}}&+5 a&-8&\frac{{\color{DeepPink}- 2 a^{2}}}{{\color{Magenta}7 a^{2}}} = {\color{DeepPink}- \frac{2}{7}}\\&-\phantom{- 2 a^{2}}&&&\\&- 2 a^{2}&+\frac{6 a}{7}&+\frac{2}{7}&{\color{DeepPink}- \frac{2}{7}} \left(7 a^{2}- 3 a-1\right) = - 2 a^{2}+\frac{6 a}{7}+\frac{2}{7}\\\hline\\&&\frac{29 a}{7}&- \frac{58}{7}&\end{array}$$

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

$$\begin{array}{r|rrr:c}&{\color{DeepPink}- \frac{2}{7}}&&&\text{Hints}\\\hline\\{\color{Magenta}7 a^{2}}- 3 a-1&{\color{DeepPink}- 2 a^{2}}&+5 a&-8&\frac{{\color{DeepPink}- 2 a^{2}}}{{\color{Magenta}7 a^{2}}} = {\color{DeepPink}- \frac{2}{7}}\\&-\phantom{- 2 a^{2}}&&&\\&- 2 a^{2}&+\frac{6 a}{7}&+\frac{2}{7}&{\color{DeepPink}- \frac{2}{7}} \left(7 a^{2}- 3 a-1\right) = - 2 a^{2}+\frac{6 a}{7}+\frac{2}{7}\\\hline\\&&\frac{29 a}{7}&- \frac{58}{7}&\end{array}$$

Therefore, $$$\frac{- 2 a^{2} + 5 a - 8}{7 a^{2} - 3 a - 1} = - \frac{2}{7} + \frac{\frac{29 a}{7} - \frac{58}{7}}{7 a^{2} - 3 a - 1}$$$.

Answer

$$$\frac{- 2 a^{2} + 5 a - 8}{7 a^{2} - 3 a - 1} = - \frac{2}{7} + \frac{\frac{29 a}{7} - \frac{58}{7}}{7 a^{2} - 3 a - 1}$$$A


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