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Divide $$$x^{2} + 4 x - 5$$$ by $$$1 - x$$$
Related calculators: Synthetic Division Calculator, Long Division Calculator
Your Input
Find $$$\frac{x^{2} + 4 x - 5}{1 - x}$$$ using long division.
Solution
Write the problem in the special format:
$$$\begin{array}{r|r}\hline\\- x+1&x^{2}+4 x-5\end{array}$$$
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: $$$\left(x^{2}+4 x-5\right) - \left(x^{2}- x\right) = 5 x-5$$$.
$$\begin{array}{r|rrr:c}&{\color{Purple}- x}&&&\\\hline\\{\color{Magenta}- x}+1&{\color{Purple}x^{2}}&+4 x&-5&\frac{{\color{Purple}x^{2}}}{{\color{Magenta}- x}} = {\color{Purple}- x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- x&&{\color{Purple}- x} \left(- x+1\right) = x^{2}- x\\\hline\\&&5 x&-5&\end{array}$$Step 2
Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{5 x}{- x} = -5$$$.
Write down the calculated result in the upper part of the table.
Multiply it by the divisor: $$$- 5 \left(- x+1\right) = 5 x-5$$$.
Subtract the remainder from the obtained result: $$$\left(5 x-5\right) - \left(5 x-5\right) = $$$.
$$\begin{array}{r|rrr:c}&- x&{\color{Red}-5}&&\\\hline\\{\color{Magenta}- x}+1&x^{2}&+4 x&-5&\\&-\phantom{x^{2}}&&&\\&x^{2}&- x&&\\\hline\\&&{\color{Red}5 x}&-5&\frac{{\color{Red}5 x}}{{\color{Magenta}- x}} = {\color{Red}-5}\\&&-\phantom{5 x}&&\\&&5 x&-5&{\color{Red}-5} \left(- x+1\right) = 5 x-5\\\hline\\&&&0&\end{array}$$Since the degree of the remainder is less than the degree of the divisor, then we are done.
The resulting table is shown once more:
$$\begin{array}{r|rrr:c}&{\color{Purple}- x}&{\color{Red}-5}&&\text{Hints}\\\hline\\{\color{Magenta}- x}+1&{\color{Purple}x^{2}}&+4 x&-5&\frac{{\color{Purple}x^{2}}}{{\color{Magenta}- x}} = {\color{Purple}- x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- x&&{\color{Purple}- x} \left(- x+1\right) = x^{2}- x\\\hline\\&&{\color{Red}5 x}&-5&\frac{{\color{Red}5 x}}{{\color{Magenta}- x}} = {\color{Red}-5}\\&&-\phantom{5 x}&&\\&&5 x&-5&{\color{Red}-5} \left(- x+1\right) = 5 x-5\\\hline\\&&&0&\end{array}$$Therefore, $$$\frac{x^{2} + 4 x - 5}{1 - x} = \left(- x - 5\right) + \frac{0}{1 - x} = - x - 5$$$.
Answer
$$$\frac{x^{2} + 4 x - 5}{1 - x} = \left(- x - 5\right) + \frac{0}{1 - x} = - x - 5$$$A