Algebra Examples

$\frac{x^{4} - 5 x^{3} + x - 4}{x^{2} - 7 x + 1}$

Step 1

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

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

Step 2

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

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 7 x^{3} + x^{2}$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

Step 5

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

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

Step 6

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

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

Step 7

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

$x^{2}$

$2 x$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 14 x^{2} + 2 x$

$x^{2}$

$2 x$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

Step 10

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

$x^{2}$

$2 x$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

Step 11

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

$x^{2}$

$2 x$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

$4$

Step 12

Divide the highest order term in the dividend $13 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$2 x$

$13$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

$4$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$13$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

$4$

$13 x^{2}$

$91 x$

$13$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{2} - 91 x + 13$

$x^{2}$

$2 x$

$13$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

$4$

$13 x^{2}$

$91 x$

$13$

Step 15

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

$x^{2}$

$2 x$

$13$

$x^{2}$

$7 x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$7 x^{3}$

$x^{2}$

$2 x^{3}$

$x^{2}$

$x$

$2 x^{3}$

$14 x^{2}$

$2 x$

$13 x^{2}$

$x$

$4$

$13 x^{2}$

$91 x$

$13$

$90 x$

$17$

Step 16

The quotient is $x^{2} + 2 x + 13$ .

$x^{2} + 2 x + 13$