Algebra Examples

$(x^{4} + 2 x^{3} + 5 x^{2} + 3 x) \div (x^{2} - x)$

Step 1

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

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

Step 2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - x^{3} + 0$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

Step 5

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

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

Step 6

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

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

Step 7

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$3 x$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 3 x^{2} + 0$

$x^{2}$

$3 x$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

Step 10

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

Step 11

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

$0$

Step 12

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$3 x$

$8$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$8$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

$0$

$8 x^{2}$

$8 x$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} - 8 x + 0$

$x^{2}$

$3 x$

$8$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

$0$

$8 x^{2}$

$8 x$

$0$

Step 15

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

$x^{2}$

$3 x$

$8$

$x^{2}$

$x$

$0$

$x^{4}$

$2 x^{3}$

$5 x^{2}$

$3 x$

$0$

$x^{4}$

$x^{3}$

$0$

$3 x^{3}$

$5 x^{2}$

$3 x$

$3 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$3 x$

$0$

$8 x^{2}$

$8 x$

$0$

$11 x$

$0$

Step 16

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

$x^{2} + 3 x + 8 + \frac{11 x}{x^{2} - x}$