Finite Math Examples

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

Step 1

Expand $6 x^{2} - 6 + 8 x^{3}$ .

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Step 1.1

Move $- 6$ .

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

Step 1.2

Reorder $6 x^{2}$ and $8 x^{3}$ .

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

Step 2

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

$4 x$

$3$

$8 x^{3}$

$6 x^{2}$

$0 x$

$6$

Step 3

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $4 x$ .

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

Step 4

Multiply the new quotient term by the divisor.

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

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} - 6 x^{2}$

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

Step 6

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

				$2 x^{2}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$

Step 7

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

				$2 x^{2}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$

Step 8

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	+	$3 x$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$

Step 9

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					+	$12 x^{2}$	-	$9 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} - 9 x$

				$2 x^{2}$	+	$3 x$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$

Step 11

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

				$2 x^{2}$	+	$3 x$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$

Step 12

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

				$2 x^{2}$	+	$3 x$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$	-	$6$

Step 13

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	+	$3 x$	+	$\frac{9}{4}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$	-	$6$

Step 14

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$	+	$\frac{9}{4}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$	-	$6$
							+	$9 x$	-	$\frac{27}{4}$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - \frac{27}{4}$

				$2 x^{2}$	+	$3 x$	+	$\frac{9}{4}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$	-	$6$
							-	$9 x$	+	$\frac{27}{4}$

Step 16

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

				$2 x^{2}$	+	$3 x$	+	$\frac{9}{4}$
$4 x$	-	$3$		$8 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$6$
			-	$8 x^{3}$	+	$6 x^{2}$
					+	$12 x^{2}$	+	$0 x$
					-	$12 x^{2}$	+	$9 x$
							+	$9 x$	-	$6$
							-	$9 x$	+	$\frac{27}{4}$
									+	$\frac{3}{4}$

Step 17

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

$2 x^{2} + 3 x + \frac{9}{4} + \frac{3}{4 (4 x - 3)}$