Pre-Algebra Examples

$\frac{6 x^{4} - 2 x^{3} - 13 x - 7}{3 x + 8}$

Step 1

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

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 16 x^{3}$

				$2 x^{3}$
$3 x$	+	$8$		$6 x^{4}$	-	$2 x^{3}$	+	$0 x^{2}$	-	$13 x$	-	$7$
			-	$6 x^{4}$	-	$16 x^{3}$

Step 5

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

				$2 x^{3}$
$3 x$	+	$8$		$6 x^{4}$	-	$2 x^{3}$	+	$0 x^{2}$	-	$13 x$	-	$7$
			-	$6 x^{4}$	-	$16 x^{3}$
					-	$18 x^{3}$

Step 6

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

$2 x^{3}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

Step 7

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$6 x^{2}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$6 x^{2}$
$3 x$	+	$8$		$6 x^{4}$	-	$2 x^{3}$	+	$0 x^{2}$	-	$13 x$	-	$7$
			-	$6 x^{4}$	-	$16 x^{3}$
					-	$18 x^{3}$	+	$0 x^{2}$
					-	$18 x^{3}$	-	$48 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} - 48 x^{2}$

$2 x^{3}$

$6 x^{2}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

Step 10

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

$2 x^{3}$

$6 x^{2}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

Step 11

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

$2 x^{3}$

$6 x^{2}$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

Step 12

Divide the highest order term in the dividend $48 x^{2}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$6 x^{2}$

$16 x$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$6 x^{2}$

$16 x$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $48 x^{2} + 128 x$

$2 x^{3}$

$6 x^{2}$

$16 x$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

Step 15

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

$2 x^{3}$

$6 x^{2}$

$16 x$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

$141 x$

Step 16

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

$2 x^{3}$

$6 x^{2}$

$16 x$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

$141 x$

$7$

Step 17

Divide the highest order term in the dividend $- 141 x$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$6 x^{2}$

$16 x$

$47$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

$141 x$

$7$

Step 18

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$47$
$3 x$	+	$8$		$6 x^{4}$	-	$2 x^{3}$	+	$0 x^{2}$	-	$13 x$	-	$7$
			-	$6 x^{4}$	-	$16 x^{3}$
					-	$18 x^{3}$	+	$0 x^{2}$
					+	$18 x^{3}$	+	$48 x^{2}$
							+	$48 x^{2}$	-	$13 x$
							-	$48 x^{2}$	-	$128 x$
									-	$141 x$	-	$7$
									-	$141 x$	-	$376$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 141 x - 376$

$2 x^{3}$

$6 x^{2}$

$16 x$

$47$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

$141 x$

$7$

$141 x$

$376$

Step 20

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

$2 x^{3}$

$6 x^{2}$

$16 x$

$47$

$3 x$

$8$

$6 x^{4}$

$2 x^{3}$

$0 x^{2}$

$13 x$

$7$

$6 x^{4}$

$16 x^{3}$

$18 x^{3}$

$0 x^{2}$

$18 x^{3}$

$48 x^{2}$

$13 x$

$48 x^{2}$

$128 x$

$141 x$

$7$

$141 x$

$376$

$369$

Step 21

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

$2 x^{3} - 6 x^{2} + 16 x - 47 + \frac{369}{3 x + 8}$