Algebra Examples

$(x^{6} + 5 x^{4} + 3 x^{2} - 2 x)$ entre $(x^{2} - x + 3)$

Step 1

Write the problem as a mathematical expression.

$\frac{(x^{6} + 5 x^{4} + 3 x^{2} - 2 x)}{(x^{2} - x + 3)}$

Step 2

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

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

Step 3

Divide the highest order term in the dividend $x^{6}$ by the highest order term in divisor $x^{2}$ .

$x^{4}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

Step 4

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} - x^{5} + 3 x^{4}$

$x^{4}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

Step 6

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

$x^{4}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

Step 7

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

$x^{4}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

Step 8

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x^{2}$ .

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

Step 9

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} - x^{4} + 3 x^{3}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

Step 11

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

Step 12

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 13

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{4}$

$x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 14

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} - 3 x^{3} + 9 x^{2}$

$x^{4}$

$x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

Step 16

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

$x^{4}$

$x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

Step 17

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

$x^{4}$

$x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

$2 x$

$0$

Step 18

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$6$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

$2 x$

$0$

Step 19

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$6$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

$2 x$

$0$

$6 x^{2}$

$6 x$

$18$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 6 x - 18$

$x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$6$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

$2 x$

$0$

$6 x^{2}$

$6 x$

$18$

Step 21

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

$x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$6$

$x^{2}$

$x$

$3$

$x^{6}$

$0 x^{5}$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{6}$

$x^{5}$

$3 x^{4}$

$x^{5}$

$2 x^{4}$

$0 x^{3}$

$x^{5}$

$x^{4}$

$3 x^{3}$

$3 x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{4}$

$3 x^{3}$

$9 x^{2}$

$6 x^{2}$

$2 x$

$0$

$6 x^{2}$

$6 x$

$18$

$8 x$

$18$

Step 22

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

$x^{4} + x^{3} + 3 x^{2} - 6 + \frac{- 8 x + 18}{x^{2} - x + 3}$