Polynomial Long Division Calculator - EMathHelp

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Polynomial Long Division Calculator

Perform the long division of polynomials step by step

The calculator will perform the long division of polynomials, with steps shown.

Related calculators: Synthetic Division Calculator, Long Division Calculator

Divide (dividend): By (divisor):

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Your Input

Find $$$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7}$$$ using long division.

Solution

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\x-7&x^{3}- 12 x^{2}+38 x-17\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{3}}{x} = x^{2}$$$.

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

Multiply it by the divisor: $$$x^{2} \left(x-7\right) = x^{3}- 7 x^{2}$$$.

Subtract the dividend from the obtained result: $$$\left(x^{3}- 12 x^{2}+38 x-17\right) - \left(x^{3}- 7 x^{2}\right) = - 5 x^{2}+38 x-17$$$.

$$\begin{array}{r|rrrr:c}&{\color{BlueViolet}x^{2}}&&&&\\\hline\\{\color{Magenta}x}-7&{\color{BlueViolet}x^{3}}&- 12 x^{2}&+38 x&-17&\frac{{\color{BlueViolet}x^{3}}}{{\color{Magenta}x}} = {\color{BlueViolet}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&{\color{BlueViolet}x^{2}} \left(x-7\right) = x^{3}- 7 x^{2}\\\hline\\&&- 5 x^{2}&+38 x&-17&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 5 x^{2}}{x} = - 5 x$$$.

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

Multiply it by the divisor: $$$- 5 x \left(x-7\right) = - 5 x^{2}+35 x$$$.

Subtract the remainder from the obtained result: $$$\left(- 5 x^{2}+38 x-17\right) - \left(- 5 x^{2}+35 x\right) = 3 x-17$$$.

$$\begin{array}{r|rrrr:c}&x^{2}&{\color{DarkBlue}- 5 x}&&&\\\hline\\{\color{Magenta}x}-7&x^{3}&- 12 x^{2}&+38 x&-17&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&\\\hline\\&&{\color{DarkBlue}- 5 x^{2}}&+38 x&-17&\frac{{\color{DarkBlue}- 5 x^{2}}}{{\color{Magenta}x}} = {\color{DarkBlue}- 5 x}\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&{\color{DarkBlue}- 5 x} \left(x-7\right) = - 5 x^{2}+35 x\\\hline\\&&&3 x&-17&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{3 x}{x} = 3$$$.

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

Multiply it by the divisor: $$$3 \left(x-7\right) = 3 x-21$$$.

Subtract the remainder from the obtained result: $$$\left(3 x-17\right) - \left(3 x-21\right) = 4$$$.

$$\begin{array}{r|rrrr:c}&x^{2}&- 5 x&{\color{Violet}+3}&&\\\hline\\{\color{Magenta}x}-7&x^{3}&- 12 x^{2}&+38 x&-17&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&\\\hline\\&&- 5 x^{2}&+38 x&-17&\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&\\\hline\\&&&{\color{Violet}3 x}&-17&\frac{{\color{Violet}3 x}}{{\color{Magenta}x}} = {\color{Violet}3}\\&&&-\phantom{3 x}&&\\&&&3 x&-21&{\color{Violet}3} \left(x-7\right) = 3 x-21\\\hline\\&&&&4&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrr:c}&{\color{BlueViolet}x^{2}}&{\color{DarkBlue}- 5 x}&{\color{Violet}+3}&&\text{Hints}\\\hline\\{\color{Magenta}x}-7&{\color{BlueViolet}x^{3}}&- 12 x^{2}&+38 x&-17&\frac{{\color{BlueViolet}x^{3}}}{{\color{Magenta}x}} = {\color{BlueViolet}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&{\color{BlueViolet}x^{2}} \left(x-7\right) = x^{3}- 7 x^{2}\\\hline\\&&{\color{DarkBlue}- 5 x^{2}}&+38 x&-17&\frac{{\color{DarkBlue}- 5 x^{2}}}{{\color{Magenta}x}} = {\color{DarkBlue}- 5 x}\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&{\color{DarkBlue}- 5 x} \left(x-7\right) = - 5 x^{2}+35 x\\\hline\\&&&{\color{Violet}3 x}&-17&\frac{{\color{Violet}3 x}}{{\color{Magenta}x}} = {\color{Violet}3}\\&&&-\phantom{3 x}&&\\&&&3 x&-21&{\color{Violet}3} \left(x-7\right) = 3 x-21\\\hline\\&&&&4&\end{array}$$

Therefore, $$$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7} = \left(x^{2} - 5 x + 3\right) + \frac{4}{x - 7}$$$.

Answer

$$$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7} = \left(x^{2} - 5 x + 3\right) + \frac{4}{x - 7}$$$A

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