Divide Using Long Polynomial Division (x^2-6x-6x^3+x^4)÷(6+x)

$(x^{2} - 6 x - 6 x^{3} + x^{4}) \div (6 + x)$

Step 1

Expand $x^{2} - 6 x - 6 x^{3} + x^{4}$ .

Move $- 6 x$ .

$\frac{x^{2} - 6 x^{3} - 6 x + x^{4}}{6 + x}$

Reorder $x^{2}$ and $- 6 x^{3}$ .

$\frac{- 6 x^{3} + x^{2} - 6 x + x^{4}}{6 + x}$

Move $- 6 x$ .

$\frac{- 6 x^{3} + x^{2} + x^{4} - 6 x}{6 + x}$

Move $x^{2}$ .

$\frac{- 6 x^{3} + x^{4} + x^{2} - 6 x}{6 + x}$

Reorder $- 6 x^{3}$ and $x^{4}$ .

$\frac{x^{4} - 6 x^{3} + x^{2} - 6 x}{6 + x}$

Step 2

Reorder $6$ and $x$ .

$\frac{x^{4} - 6 x^{3} + x^{2} - 6 x}{x + 6}$

Step 3

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

Step 4

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

Step 5

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

Step 6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 6 x^{3}$

$x^{3}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

Step 7

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

Step 8

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

Step 9

Divide the highest order term in the dividend $- 12 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

Step 10

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

Step 11

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{3} - 72 x^{2}$

$x^{3}$

$12 x^{2}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

Step 12

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

Step 13

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

Step 14

Divide the highest order term in the dividend $73 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$73 x$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

Step 15

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$73 x$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

Step 16

The expression needs to be subtracted from the dividend, so change all the signs in $73 x^{2} + 438 x$

$x^{3}$

$12 x^{2}$

$73 x$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

Step 17

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$73 x$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

Step 18

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$73 x$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

$0$

Step 19

Divide the highest order term in the dividend $- 444 x$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$73 x$

$444$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

$0$

Step 20

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$73 x$

$444$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

$0$

$444 x$

$2664$

Step 21

The expression needs to be subtracted from the dividend, so change all the signs in $- 444 x - 2664$

$x^{3}$

$12 x^{2}$

$73 x$

$444$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

$0$

$444 x$

$2664$

Step 22

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$73 x$

$444$

$x$

$6$

$x^{4}$

$6 x^{3}$

$x^{2}$

$6 x$

$0$

$x^{4}$

$6 x^{3}$

$12 x^{3}$

$x^{2}$

$12 x^{3}$

$72 x^{2}$

$73 x^{2}$

$6 x$

$73 x^{2}$

$438 x$

$444 x$

$0$

$444 x$

$2664$

Step 23

The final answer is the quotient plus the remainder over the divisor.

$x^{3} - 12 x^{2} + 73 x - 444 + \frac{2664}{x + 6}$

Divide Using Long Polynomial Division (x^2-6x-6x^3+x^4)÷(6+x)

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