Divide x3+7x2+1x^{3} + 7 x^{2} + 1 by x1x - 1

The calculator will divide x3+7x2+1x^{3} + 7 x^{2} + 1 by x1x - 1 using long division, with steps shown.

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Your Input

Find x3+7x2+1x1\frac{x^{3} + 7 x^{2} + 1}{x - 1} using long division.

Solution

Write the problem in the special format (missed terms are written with zero coefficients):

x1x3+7x2+0x+1\begin{array}{r|r}\hline\\x-1&x^{3}+7 x^{2}+0 x+1\end{array}

Step 1

Divide the leading term of the dividend by the leading term of the divisor: x3x=x2\frac{x^{3}}{x} = x^{2}.

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

Multiply it by the divisor: x2(x1)=x3x2x^{2} \left(x-1\right) = x^{3}- x^{2}.

Subtract the dividend from the obtained result: (x3+7x2+1)(x3x2)=8x2+1\left(x^{3}+7 x^{2}+1\right) - \left(x^{3}- x^{2}\right) = 8 x^{2}+1.

x2x1x3+7x2+0x+1x3x=x2x3x3x2x2(x1)=x3x28x2+0x+1\begin{array}{r|rrrr:c}&{\color{DarkBlue}x^{2}}&&&&\\\hline\\{\color{Magenta}x}-1&{\color{DarkBlue}x^{3}}&+7 x^{2}&+0 x&+1&\frac{{\color{DarkBlue}x^{3}}}{{\color{Magenta}x}} = {\color{DarkBlue}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- x^{2}&&&{\color{DarkBlue}x^{2}} \left(x-1\right) = x^{3}- x^{2}\\\hline\\&&8 x^{2}&+0 x&+1&\end{array}

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: 8x2x=8x\frac{8 x^{2}}{x} = 8 x.

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

Multiply it by the divisor: 8x(x1)=8x28x8 x \left(x-1\right) = 8 x^{2}- 8 x.

Subtract the remainder from the obtained result: (8x2+1)(8x28x)=8x+1\left(8 x^{2}+1\right) - \left(8 x^{2}- 8 x\right) = 8 x+1.

x2+8xx1x3+7x2+0x+1x3x3x28x2+0x+18x2x=8x8x28x28x8x(x1)=8x28x8x+1\begin{array}{r|rrrr:c}&x^{2}&{\color{Green}+8 x}&&&\\\hline\\{\color{Magenta}x}-1&x^{3}&+7 x^{2}&+0 x&+1&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- x^{2}&&&\\\hline\\&&{\color{Green}8 x^{2}}&+0 x&+1&\frac{{\color{Green}8 x^{2}}}{{\color{Magenta}x}} = {\color{Green}8 x}\\&&-\phantom{8 x^{2}}&&&\\&&8 x^{2}&- 8 x&&{\color{Green}8 x} \left(x-1\right) = 8 x^{2}- 8 x\\\hline\\&&&8 x&+1&\end{array}

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: 8xx=8\frac{8 x}{x} = 8.

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

Multiply it by the divisor: 8(x1)=8x88 \left(x-1\right) = 8 x-8.

Subtract the remainder from the obtained result: (8x+1)(8x8)=9\left(8 x+1\right) - \left(8 x-8\right) = 9.

x2+8x+8x1x3+7x2+0x+1x3x3x28x2+0x+18x28x28x8x+18xx=88x8x88(x1)=8x89\begin{array}{r|rrrr:c}&x^{2}&+8 x&{\color{Chartreuse}+8}&&\\\hline\\{\color{Magenta}x}-1&x^{3}&+7 x^{2}&+0 x&+1&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- x^{2}&&&\\\hline\\&&8 x^{2}&+0 x&+1&\\&&-\phantom{8 x^{2}}&&&\\&&8 x^{2}&- 8 x&&\\\hline\\&&&{\color{Chartreuse}8 x}&+1&\frac{{\color{Chartreuse}8 x}}{{\color{Magenta}x}} = {\color{Chartreuse}8}\\&&&-\phantom{8 x}&&\\&&&8 x&-8&{\color{Chartreuse}8} \left(x-1\right) = 8 x-8\\\hline\\&&&&9&\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:

x2+8x+8Hintsx1x3+7x2+0x+1x3x=x2x3x3x2x2(x1)=x3x28x2+0x+18x2x=8x8x28x28x8x(x1)=8x28x8x+18xx=88x8x88(x1)=8x89\begin{array}{r|rrrr:c}&{\color{DarkBlue}x^{2}}&{\color{Green}+8 x}&{\color{Chartreuse}+8}&&\text{Hints}\\\hline\\{\color{Magenta}x}-1&{\color{DarkBlue}x^{3}}&+7 x^{2}&+0 x&+1&\frac{{\color{DarkBlue}x^{3}}}{{\color{Magenta}x}} = {\color{DarkBlue}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- x^{2}&&&{\color{DarkBlue}x^{2}} \left(x-1\right) = x^{3}- x^{2}\\\hline\\&&{\color{Green}8 x^{2}}&+0 x&+1&\frac{{\color{Green}8 x^{2}}}{{\color{Magenta}x}} = {\color{Green}8 x}\\&&-\phantom{8 x^{2}}&&&\\&&8 x^{2}&- 8 x&&{\color{Green}8 x} \left(x-1\right) = 8 x^{2}- 8 x\\\hline\\&&&{\color{Chartreuse}8 x}&+1&\frac{{\color{Chartreuse}8 x}}{{\color{Magenta}x}} = {\color{Chartreuse}8}\\&&&-\phantom{8 x}&&\\&&&8 x&-8&{\color{Chartreuse}8} \left(x-1\right) = 8 x-8\\\hline\\&&&&9&\end{array}

Therefore, x3+7x2+1x1=(x2+8x+8)+9x1\frac{x^{3} + 7 x^{2} + 1}{x - 1} = \left(x^{2} + 8 x + 8\right) + \frac{9}{x - 1}.

Answer

x3+7x2+1x1=(x2+8x+8)+9x1\frac{x^{3} + 7 x^{2} + 1}{x - 1} = \left(x^{2} + 8 x + 8\right) + \frac{9}{x - 1}A