Pre-Algebra Examples

$\frac{4 b^{4} + 2 b^{3} - 25 b^{2} - 2 b + 8}{b^{2} - 2}$

Step 1

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

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

Step 2

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

$4 b^{2}$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

Step 3

Multiply the new quotient term by the divisor.

$4 b^{2}$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $4 b^{4} + 0 - 8 b^{2}$

$4 b^{2}$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

Step 5

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

$4 b^{2}$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

Step 6

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

$4 b^{2}$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

Step 7

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

$4 b^{2}$

$2 b$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

Step 8

Multiply the new quotient term by the divisor.

$4 b^{2}$

$2 b$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

$2 b^{3}$

$0$

$4 b$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $2 b^{3} + 0 - 4 b$

$4 b^{2}$

$2 b$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

$2 b^{3}$

$0$

$4 b$

Step 10

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

$4 b^{2}$

$2 b$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

$2 b^{3}$

$0$

$4 b$

$17 b^{2}$

$2 b$

Step 11

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

$4 b^{2}$

$2 b$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

$2 b^{3}$

$0$

$4 b$

$17 b^{2}$

$2 b$

$8$

Step 12

Divide the highest order term in the dividend $- 17 b^{2}$ by the highest order term in divisor $b^{2}$ .

$4 b^{2}$

$2 b$

$17$

$b^{2}$

$0 b$

$2$

$4 b^{4}$

$2 b^{3}$

$25 b^{2}$

$2 b$

$8$

$4 b^{4}$

$0$

$8 b^{2}$

$2 b^{3}$

$17 b^{2}$

$2 b$

$2 b^{3}$

$0$

$4 b$

$17 b^{2}$

$2 b$

$8$

Step 13

Multiply the new quotient term by the divisor.

						$4 b^{2}$	+	$2 b$	-	$17$
$b^{2}$	+	$0 b$	-	$2$		$4 b^{4}$	+	$2 b^{3}$	-	$25 b^{2}$	-	$2 b$	+	$8$
					-	$4 b^{4}$	-	$0$	+	$8 b^{2}$
							+	$2 b^{3}$	-	$17 b^{2}$	-	$2 b$
							-	$2 b^{3}$	-	$0$	+	$4 b$
									-	$17 b^{2}$	+	$2 b$	+	$8$
									-	$17 b^{2}$	+	$0$	+	$34$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 b^{2} + 0 + 34$

						$4 b^{2}$	+	$2 b$	-	$17$
$b^{2}$	+	$0 b$	-	$2$		$4 b^{4}$	+	$2 b^{3}$	-	$25 b^{2}$	-	$2 b$	+	$8$
					-	$4 b^{4}$	-	$0$	+	$8 b^{2}$
							+	$2 b^{3}$	-	$17 b^{2}$	-	$2 b$
							-	$2 b^{3}$	-	$0$	+	$4 b$
									-	$17 b^{2}$	+	$2 b$	+	$8$
									+	$17 b^{2}$	-	$0$	-	$34$

Step 15

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

						$4 b^{2}$	+	$2 b$	-	$17$
$b^{2}$	+	$0 b$	-	$2$		$4 b^{4}$	+	$2 b^{3}$	-	$25 b^{2}$	-	$2 b$	+	$8$
					-	$4 b^{4}$	-	$0$	+	$8 b^{2}$
							+	$2 b^{3}$	-	$17 b^{2}$	-	$2 b$
							-	$2 b^{3}$	-	$0$	+	$4 b$
									-	$17 b^{2}$	+	$2 b$	+	$8$
									+	$17 b^{2}$	-	$0$	-	$34$
											+	$2 b$	-	$26$

Step 16

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

$4 b^{2} + 2 b - 17 + \frac{2 b - 26}{b^{2} - 2}$