Divide $5 x^{9} - 4 x^{2} + 2$ by $5 x + 10$

Find $\frac{5 x^{9} - 4 x^{2} + 2}{5 x + 10}$ using long division.

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

$\begin{array}{r|r}\hline\\5 x+10&5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2\end{array}$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $\frac{5 x^{9}}{5 x} = x^{8}$.

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

Multiply it by the divisor: $x^{8} \left(5 x+10\right) = 5 x^{9}+10 x^{8}$.

Subtract the dividend from the obtained result: $\left(5 x^{9}- 4 x^{2}+2\right) - \left(5 x^{9}+10 x^{8}\right) = - 10 x^{8}- 4 x^{2}+2$.

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 10 x^{8}}{5 x} = - 2 x^{7}$.

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

Multiply it by the divisor: $- 2 x^{7} \left(5 x+10\right) = - 10 x^{8}- 20 x^{7}$.

Subtract the remainder from the obtained result: $\left(- 10 x^{8}- 4 x^{2}+2\right) - \left(- 10 x^{8}- 20 x^{7}\right) = 20 x^{7}- 4 x^{2}+2$.

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{20 x^{7}}{5 x} = 4 x^{6}$.

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

Multiply it by the divisor: $4 x^{6} \left(5 x+10\right) = 20 x^{7}+40 x^{6}$.

Subtract the remainder from the obtained result: $\left(20 x^{7}- 4 x^{2}+2\right) - \left(20 x^{7}+40 x^{6}\right) = - 40 x^{6}- 4 x^{2}+2$.

Step 4

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 40 x^{6}}{5 x} = - 8 x^{5}$.

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

Multiply it by the divisor: $- 8 x^{5} \left(5 x+10\right) = - 40 x^{6}- 80 x^{5}$.

Subtract the remainder from the obtained result: $\left(- 40 x^{6}- 4 x^{2}+2\right) - \left(- 40 x^{6}- 80 x^{5}\right) = 80 x^{5}- 4 x^{2}+2$.

Step 5

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{80 x^{5}}{5 x} = 16 x^{4}$.

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

Multiply it by the divisor: $16 x^{4} \left(5 x+10\right) = 80 x^{5}+160 x^{4}$.

Subtract the remainder from the obtained result: $\left(80 x^{5}- 4 x^{2}+2\right) - \left(80 x^{5}+160 x^{4}\right) = - 160 x^{4}- 4 x^{2}+2$.

Step 6

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 160 x^{4}}{5 x} = - 32 x^{3}$.

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

Multiply it by the divisor: $- 32 x^{3} \left(5 x+10\right) = - 160 x^{4}- 320 x^{3}$.

Subtract the remainder from the obtained result: $\left(- 160 x^{4}- 4 x^{2}+2\right) - \left(- 160 x^{4}- 320 x^{3}\right) = 320 x^{3}- 4 x^{2}+2$.

Step 7

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{320 x^{3}}{5 x} = 64 x^{2}$.

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

Multiply it by the divisor: $64 x^{2} \left(5 x+10\right) = 320 x^{3}+640 x^{2}$.

Subtract the remainder from the obtained result: $\left(320 x^{3}- 4 x^{2}+2\right) - \left(320 x^{3}+640 x^{2}\right) = - 644 x^{2}+2$.

Step 8

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 644 x^{2}}{5 x} = - \frac{644 x}{5}$.

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

Multiply it by the divisor: $- \frac{644 x}{5} \left(5 x+10\right) = - 644 x^{2}- 1288 x$.

Subtract the remainder from the obtained result: $\left(- 644 x^{2}+2\right) - \left(- 644 x^{2}- 1288 x\right) = 1288 x+2$.