Finite Math Examples

$\frac{- 2 - 18 x^{4} - 10 x^{2} + 14 x}{3 x^{2} - 2 x - 1}$

Step 1

Expand $- 2 - 18 x^{4} - 10 x^{2} + 14 x$ .

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Step 1.1

Reorder $- 2$ and $- 18 x^{4}$ .

$\frac{- 18 x^{4} - 2 - 10 x^{2} + 14 x}{3 x^{2} - 2 x - 1}$

Step 1.2

Move $- 2$ .

$\frac{- 18 x^{4} - 10 x^{2} - 2 + 14 x}{3 x^{2} - 2 x - 1}$

Step 1.3

Move $- 2$ .

$\frac{- 18 x^{4} - 10 x^{2} + 14 x - 2}{3 x^{2} - 2 x - 1}$

Step 2

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

$3 x^{2}$

$2 x$

$1$

$18 x^{4}$

$0 x^{3}$

$10 x^{2}$

$14 x$

$2$

Step 3

Divide the highest order term in the dividend $- 18 x^{4}$ by the highest order term in divisor $3 x^{2}$ .

						-	$6 x^{2}$
	$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$

Step 4

Multiply the new quotient term by the divisor.

$6 x^{2}$

$3 x^{2}$

$2 x$

$1$

$18 x^{4}$

$0 x^{3}$

$10 x^{2}$

$14 x$

$2$

$18 x^{4}$

$12 x^{3}$

$6 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{4} + 12 x^{3} + 6 x^{2}$

					-	$6 x^{2}$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$

Step 6

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

					-	$6 x^{2}$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$

Step 7

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

					-	$6 x^{2}$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$

Step 8

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

					-	$6 x^{2}$	-	$4 x$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$

Step 9

Multiply the new quotient term by the divisor.

$6 x^{2}$

$4 x$

$3 x^{2}$

$2 x$

$1$

$18 x^{4}$

$0 x^{3}$

$10 x^{2}$

$14 x$

$2$

$18 x^{4}$

$12 x^{3}$

$6 x^{2}$

$12 x^{3}$

$16 x^{2}$

$14 x$

$12 x^{3}$

$8 x^{2}$

$4 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{3} + 8 x^{2} + 4 x$

					-	$6 x^{2}$	-	$4 x$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$

Step 11

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

$6 x^{2}$

$4 x$

$3 x^{2}$

$2 x$

$1$

$18 x^{4}$

$0 x^{3}$

$10 x^{2}$

$14 x$

$2$

$18 x^{4}$

$12 x^{3}$

$6 x^{2}$

$12 x^{3}$

$16 x^{2}$

$14 x$

$12 x^{3}$

$8 x^{2}$

$4 x$

$24 x^{2}$

$10 x$

Step 12

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

					-	$6 x^{2}$	-	$4 x$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$
									-	$24 x^{2}$	+	$10 x$	-	$2$

Step 13

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

					-	$6 x^{2}$	-	$4 x$	-	$8$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$
									-	$24 x^{2}$	+	$10 x$	-	$2$

Step 14

Multiply the new quotient term by the divisor.

					-	$6 x^{2}$	-	$4 x$	-	$8$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$
									-	$24 x^{2}$	+	$10 x$	-	$2$
									-	$24 x^{2}$	+	$16 x$	+	$8$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} + 16 x + 8$

					-	$6 x^{2}$	-	$4 x$	-	$8$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$
									-	$24 x^{2}$	+	$10 x$	-	$2$
									+	$24 x^{2}$	-	$16 x$	-	$8$

Step 16

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

					-	$6 x^{2}$	-	$4 x$	-	$8$
$3 x^{2}$	-	$2 x$	-	$1$	-	$18 x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	+	$14 x$	-	$2$
					+	$18 x^{4}$	-	$12 x^{3}$	-	$6 x^{2}$
							-	$12 x^{3}$	-	$16 x^{2}$	+	$14 x$
							+	$12 x^{3}$	-	$8 x^{2}$	-	$4 x$
									-	$24 x^{2}$	+	$10 x$	-	$2$
									+	$24 x^{2}$	-	$16 x$	-	$8$
											-	$6 x$	-	$10$

Step 17

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

$- 6 x^{2} - 4 x - 8 + \frac{- 6 x - 10}{3 x^{2} - 2 x - 1}$