Pre-Algebra Examples

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

Step 1

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

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

Step 2

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

Step 4

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

Step 5

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

Step 6

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 7

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 9

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 10

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

Step 11

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

Step 12

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

Step 13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} + 36 x$

$x^{3}$

$3 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

Step 15

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

$x^{3}$

$3 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

Step 16

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

$x^{3}$

$3 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

$4$

Step 17

Divide the highest order term in the dividend $- 35 x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$12 x$

$35$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

$4$

Step 18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$12 x$

$35$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

$4$

$35 x$

$105$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x - 105$

$x^{3}$

$3 x^{2}$

$12 x$

$35$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

$4$

$35 x$

$105$

Step 20

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

$x^{3}$

$3 x^{2}$

$12 x$

$35$

$x$

$3$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$4$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$3 x^{3}$

$9 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$35 x$

$4$

$35 x$

$105$

$109$

Step 21

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

$x^{3} - 3 x^{2} + 12 x - 35 + \frac{109}{x + 3}$