Pre-Algebra Examples

$(x^{4} + 13 x^{3} - 46 x^{2} + 10 x - 48) \div (x - 3)$

Step 1

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

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

Step 2

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

$x^{3}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

Step 4

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

$x^{3}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

Step 5

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

$x^{3}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

Step 6

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

$x^{3}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

Step 7

Divide the highest order term in the dividend $16 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$16 x^{2}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$16 x^{2}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{3} - 48 x^{2}$

$x^{3}$

$16 x^{2}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

Step 10

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

$x^{3}$

$16 x^{2}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

Step 11

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

$x^{3}$

$16 x^{2}$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

Step 12

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

$x^{3}$

$16 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

Step 13

Multiply the new quotient term by the divisor.

$x^{3}$

$16 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

Step 14

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

$x^{3}$

$16 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

Step 15

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

$x^{3}$

$16 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

Step 16

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

$x^{3}$

$16 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

$48$

Step 17

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

$x^{3}$

$16 x^{2}$

$2 x$

$16$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

$48$

Step 18

Multiply the new quotient term by the divisor.

$x^{3}$

$16 x^{2}$

$2 x$

$16$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

$48$

$16 x$

$48$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 48$

$x^{3}$

$16 x^{2}$

$2 x$

$16$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

$48$

$16 x$

$48$

Step 20

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

$x^{3}$

$16 x^{2}$

$2 x$

$16$

$x$

$3$

$x^{4}$

$13 x^{3}$

$46 x^{2}$

$10 x$

$48$

$x^{4}$

$3 x^{3}$

$16 x^{3}$

$46 x^{2}$

$16 x^{3}$

$48 x^{2}$

$2 x^{2}$

$10 x$

$2 x^{2}$

$6 x$

$16 x$

$48$

$16 x$

$48$

$0$

Step 21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + 16 x^{2} + 2 x + 16$