Pre-Algebra Examples

$(8 x^{4} - 14 x^{3} - 29 x^{3} + 56 x - 12) \div (x - 3)$

Step 1

Subtract $29 x^{3}$ from $- 14 x^{3}$ .

$\frac{8 x^{4} - 43 x^{3} + 56 x - 12}{x - 3}$

Step 2

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

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

Step 3

Divide the highest order term in the dividend $8 x^{4}$ by the highest order term in divisor $x$ .

$8 x^{3}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

Step 4

Multiply the new quotient term by the divisor.

$8 x^{3}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{4} - 24 x^{3}$

$8 x^{3}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

Step 6

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

$8 x^{3}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

Step 7

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

$8 x^{3}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

Step 8

Divide the highest order term in the dividend $- 19 x^{3}$ by the highest order term in divisor $x$ .

$8 x^{3}$

$19 x^{2}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$8 x^{3}$

$19 x^{2}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 19 x^{3} + 57 x^{2}$

$8 x^{3}$

$19 x^{2}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

Step 11

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

$8 x^{3}$

$19 x^{2}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

Step 12

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

$8 x^{3}$

$19 x^{2}$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

Step 13

Divide the highest order term in the dividend $- 57 x^{2}$ by the highest order term in divisor $x$ .

$8 x^{3}$

$19 x^{2}$

$57 x$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

Step 14

Multiply the new quotient term by the divisor.

$8 x^{3}$

$19 x^{2}$

$57 x$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

$57 x^{2}$

$171 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 57 x^{2} + 171 x$

$8 x^{3}$

$19 x^{2}$

$57 x$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

$57 x^{2}$

$171 x$

Step 16

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

$8 x^{3}$

$19 x^{2}$

$57 x$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

$57 x^{2}$

$171 x$

$115 x$

Step 17

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

$8 x^{3}$

$19 x^{2}$

$57 x$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

$57 x^{2}$

$171 x$

$115 x$

$12$

Step 18

Divide the highest order term in the dividend $- 115 x$ by the highest order term in divisor $x$ .

$8 x^{3}$

$19 x^{2}$

$57 x$

$115$

$x$

$3$

$8 x^{4}$

$43 x^{3}$

$0 x^{2}$

$56 x$

$12$

$8 x^{4}$

$24 x^{3}$

$19 x^{3}$

$0 x^{2}$

$19 x^{3}$

$57 x^{2}$

$56 x$

$57 x^{2}$

$171 x$

$115 x$

$12$

Step 19

Multiply the new quotient term by the divisor.

				$8 x^{3}$	-	$19 x^{2}$	-	$57 x$	-	$115$
$x$	-	$3$		$8 x^{4}$	-	$43 x^{3}$	+	$0 x^{2}$	+	$56 x$	-	$12$
			-	$8 x^{4}$	+	$24 x^{3}$
					-	$19 x^{3}$	+	$0 x^{2}$
					+	$19 x^{3}$	-	$57 x^{2}$
							-	$57 x^{2}$	+	$56 x$
							+	$57 x^{2}$	-	$171 x$
									-	$115 x$	-	$12$
									-	$115 x$	+	$345$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 115 x + 345$

				$8 x^{3}$	-	$19 x^{2}$	-	$57 x$	-	$115$
$x$	-	$3$		$8 x^{4}$	-	$43 x^{3}$	+	$0 x^{2}$	+	$56 x$	-	$12$
			-	$8 x^{4}$	+	$24 x^{3}$
					-	$19 x^{3}$	+	$0 x^{2}$
					+	$19 x^{3}$	-	$57 x^{2}$
							-	$57 x^{2}$	+	$56 x$
							+	$57 x^{2}$	-	$171 x$
									-	$115 x$	-	$12$
									+	$115 x$	-	$345$

Step 21

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

				$8 x^{3}$	-	$19 x^{2}$	-	$57 x$	-	$115$
$x$	-	$3$		$8 x^{4}$	-	$43 x^{3}$	+	$0 x^{2}$	+	$56 x$	-	$12$
			-	$8 x^{4}$	+	$24 x^{3}$
					-	$19 x^{3}$	+	$0 x^{2}$
					+	$19 x^{3}$	-	$57 x^{2}$
							-	$57 x^{2}$	+	$56 x$
							+	$57 x^{2}$	-	$171 x$
									-	$115 x$	-	$12$
									+	$115 x$	-	$345$
											-	$357$

Step 22

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

$8 x^{3} - 19 x^{2} - 57 x - 115 - \frac{357}{x - 3}$