Pre-Algebra Examples

$\frac{20 k^{4} + 14 k^{3} + 7 k - 5}{2 k^{2} + 1}$

Step 1

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

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

Step 2

Divide the highest order term in the dividend $20 k^{4}$ by the highest order term in divisor $2 k^{2}$ .

$10 k^{2}$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

Step 3

Multiply the new quotient term by the divisor.

$10 k^{2}$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $20 k^{4} + 0 + 10 k^{2}$

$10 k^{2}$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

Step 5

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

$10 k^{2}$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

Step 6

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

$10 k^{2}$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

Step 7

Divide the highest order term in the dividend $14 k^{3}$ by the highest order term in divisor $2 k^{2}$ .

$10 k^{2}$

$7 k$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

Step 8

Multiply the new quotient term by the divisor.

$10 k^{2}$

$7 k$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

$14 k^{3}$

$0$

$7 k$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $14 k^{3} + 0 + 7 k$

$10 k^{2}$

$7 k$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

$14 k^{3}$

$0$

$7 k$

Step 10

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

$10 k^{2}$

$7 k$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

$14 k^{3}$

$0$

$7 k$

$10 k^{2}$

$0$

Step 11

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

$10 k^{2}$

$7 k$

$2 k^{2}$

$0 k$

$1$

$20 k^{4}$

$14 k^{3}$

$0 k^{2}$

$7 k$

$5$

$20 k^{4}$

$0$

$10 k^{2}$

$14 k^{3}$

$10 k^{2}$

$7 k$

$14 k^{3}$

$0$

$7 k$

$10 k^{2}$

$0$

$5$

Step 12

Divide the highest order term in the dividend $- 10 k^{2}$ by the highest order term in divisor $2 k^{2}$ .

						$10 k^{2}$	+	$7 k$	-	$5$
$2 k^{2}$	+	$0 k$	+	$1$		$20 k^{4}$	+	$14 k^{3}$	+	$0 k^{2}$	+	$7 k$	-	$5$
					-	$20 k^{4}$	-	$0$	-	$10 k^{2}$
							+	$14 k^{3}$	-	$10 k^{2}$	+	$7 k$
							-	$14 k^{3}$	-	$0$	-	$7 k$
									-	$10 k^{2}$	+	$0$	-	$5$

Step 13

Multiply the new quotient term by the divisor.

						$10 k^{2}$	+	$7 k$	-	$5$
$2 k^{2}$	+	$0 k$	+	$1$		$20 k^{4}$	+	$14 k^{3}$	+	$0 k^{2}$	+	$7 k$	-	$5$
					-	$20 k^{4}$	-	$0$	-	$10 k^{2}$
							+	$14 k^{3}$	-	$10 k^{2}$	+	$7 k$
							-	$14 k^{3}$	-	$0$	-	$7 k$
									-	$10 k^{2}$	+	$0$	-	$5$
									-	$10 k^{2}$	+	$0$	-	$5$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 k^{2} + 0 - 5$

						$10 k^{2}$	+	$7 k$	-	$5$
$2 k^{2}$	+	$0 k$	+	$1$		$20 k^{4}$	+	$14 k^{3}$	+	$0 k^{2}$	+	$7 k$	-	$5$
					-	$20 k^{4}$	-	$0$	-	$10 k^{2}$
							+	$14 k^{3}$	-	$10 k^{2}$	+	$7 k$
							-	$14 k^{3}$	-	$0$	-	$7 k$
									-	$10 k^{2}$	+	$0$	-	$5$
									+	$10 k^{2}$	-	$0$	+	$5$

Step 15

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

						$10 k^{2}$	+	$7 k$	-	$5$
$2 k^{2}$	+	$0 k$	+	$1$		$20 k^{4}$	+	$14 k^{3}$	+	$0 k^{2}$	+	$7 k$	-	$5$
					-	$20 k^{4}$	-	$0$	-	$10 k^{2}$
							+	$14 k^{3}$	-	$10 k^{2}$	+	$7 k$
							-	$14 k^{3}$	-	$0$	-	$7 k$
									-	$10 k^{2}$	+	$0$	-	$5$
									+	$10 k^{2}$	-	$0$	+	$5$
														$0$

Step 16

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

$10 k^{2} + 7 k - 5$