Finite Math Examples

$\frac{6 x^{4} + 12 x^{3} - 10 x^{2}}{3 x^{2} + 1}$

Step 1

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

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $3 x^{2}$ .

$2 x^{2}$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 0 + 2 x^{2}$

$2 x^{2}$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

Step 5

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

$2 x^{2}$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

Step 6

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

$2 x^{2}$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$2 x^{2}$

$4 x$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$4 x$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

$12 x^{3}$

$0$

$4 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 0 + 4 x$

$2 x^{2}$

$4 x$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

$12 x^{3}$

$0$

$4 x$

Step 10

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

$2 x^{2}$

$4 x$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

$12 x^{3}$

$0$

$4 x$

$12 x^{2}$

$4 x$

Step 11

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

$2 x^{2}$

$4 x$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

$12 x^{3}$

$0$

$4 x$

$12 x^{2}$

$4 x$

$0$

Step 12

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

$2 x^{2}$

$4 x$

$4$

$3 x^{2}$

$0 x$

$1$

$6 x^{4}$

$12 x^{3}$

$10 x^{2}$

$0 x$

$0$

$6 x^{4}$

$0$

$2 x^{2}$

$12 x^{3}$

$12 x^{2}$

$0 x$

$12 x^{3}$

$0$

$4 x$

$12 x^{2}$

$4 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

						$2 x^{2}$	+	$4 x$	-	$4$
$3 x^{2}$	+	$0 x$	+	$1$		$6 x^{4}$	+	$12 x^{3}$	-	$10 x^{2}$	+	$0 x$	+	$0$
					-	$6 x^{4}$	-	$0$	-	$2 x^{2}$
							+	$12 x^{3}$	-	$12 x^{2}$	+	$0 x$
							-	$12 x^{3}$	-	$0$	-	$4 x$
									-	$12 x^{2}$	-	$4 x$	+	$0$
									-	$12 x^{2}$	+	$0$	-	$4$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} + 0 - 4$

						$2 x^{2}$	+	$4 x$	-	$4$
$3 x^{2}$	+	$0 x$	+	$1$		$6 x^{4}$	+	$12 x^{3}$	-	$10 x^{2}$	+	$0 x$	+	$0$
					-	$6 x^{4}$	-	$0$	-	$2 x^{2}$
							+	$12 x^{3}$	-	$12 x^{2}$	+	$0 x$
							-	$12 x^{3}$	-	$0$	-	$4 x$
									-	$12 x^{2}$	-	$4 x$	+	$0$
									+	$12 x^{2}$	-	$0$	+	$4$

Step 15

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

						$2 x^{2}$	+	$4 x$	-	$4$
$3 x^{2}$	+	$0 x$	+	$1$		$6 x^{4}$	+	$12 x^{3}$	-	$10 x^{2}$	+	$0 x$	+	$0$
					-	$6 x^{4}$	-	$0$	-	$2 x^{2}$
							+	$12 x^{3}$	-	$12 x^{2}$	+	$0 x$
							-	$12 x^{3}$	-	$0$	-	$4 x$
									-	$12 x^{2}$	-	$4 x$	+	$0$
									+	$12 x^{2}$	-	$0$	+	$4$
											-	$4 x$	+	$4$

Step 16

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

$2 x^{2} + 4 x - 4 + \frac{- 4 x + 4}{3 x^{2} + 1}$