|
|
|
|
|
|
|
|
|
|
|
|
Divide $$$x^{2} - x$$$ by $$$x + 1$$$
Related calculators: Synthetic Division Calculator, Long Division Calculator
Your Input
Find $$$\frac{x^{2} - x}{x + 1}$$$ using long division.
Solution
Write the problem in the special format (missed terms are written with zero coefficients):
$$$\begin{array}{r|r}\hline\\x+1&x^{2}- x+0\end{array}$$$
Step 1
Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{2}}{x} = x$$$.
Write down the calculated result in the upper part of the table.
Multiply it by the divisor: $$$x \left(x+1\right) = x^{2}+x$$$.
Subtract the dividend from the obtained result: $$$\left(x^{2}- x\right) - \left(x^{2}+x\right) = - 2 x$$$.
$$\begin{array}{r|rrr:c}&{\color{Peru}x}&&&\\\hline\\{\color{Magenta}x}+1&{\color{Peru}x^{2}}&- x&+0&\frac{{\color{Peru}x^{2}}}{{\color{Magenta}x}} = {\color{Peru}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&+x&&{\color{Peru}x} \left(x+1\right) = x^{2}+x\\\hline\\&&- 2 x&+0&\end{array}$$Step 2
Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 2 x}{x} = -2$$$.
Write down the calculated result in the upper part of the table.
Multiply it by the divisor: $$$- 2 \left(x+1\right) = - 2 x-2$$$.
Subtract the remainder from the obtained result: $$$\left(- 2 x\right) - \left(- 2 x-2\right) = 2$$$.
$$\begin{array}{r|rrr:c}&x&{\color{Brown}-2}&&\\\hline\\{\color{Magenta}x}+1&x^{2}&- x&+0&\\&-\phantom{x^{2}}&&&\\&x^{2}&+x&&\\\hline\\&&{\color{Brown}- 2 x}&+0&\frac{{\color{Brown}- 2 x}}{{\color{Magenta}x}} = {\color{Brown}-2}\\&&-\phantom{- 2 x}&&\\&&- 2 x&-2&{\color{Brown}-2} \left(x+1\right) = - 2 x-2\\\hline\\&&&2&\end{array}$$Since the degree of the remainder is less than the degree of the divisor, we are done.
The resulting table is shown once more:
$$\begin{array}{r|rrr:c}&{\color{Peru}x}&{\color{Brown}-2}&&\text{Hints}\\\hline\\{\color{Magenta}x}+1&{\color{Peru}x^{2}}&- x&+0&\frac{{\color{Peru}x^{2}}}{{\color{Magenta}x}} = {\color{Peru}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&+x&&{\color{Peru}x} \left(x+1\right) = x^{2}+x\\\hline\\&&{\color{Brown}- 2 x}&+0&\frac{{\color{Brown}- 2 x}}{{\color{Magenta}x}} = {\color{Brown}-2}\\&&-\phantom{- 2 x}&&\\&&- 2 x&-2&{\color{Brown}-2} \left(x+1\right) = - 2 x-2\\\hline\\&&&2&\end{array}$$Therefore, $$$\frac{x^{2} - x}{x + 1} = \left(x - 2\right) + \frac{2}{x + 1}$$$.
Answer
$$$\frac{x^{2} - x}{x + 1} = \left(x - 2\right) + \frac{2}{x + 1}$$$A