Finite Math Examples

$\frac{x^{4} - 9 x^{3} + 95 x^{2} - 729 x + 1134}{x^{2} + 81}$

Step 1

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

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

Step 2

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

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + 81 x^{2}$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

Step 5

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

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

Step 6

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

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

Step 7

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$9 x$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

Step 8

Multiply the new quotient term by the divisor.

						$x^{2}$	-	$9 x$
$x^{2}$	+	$0 x$	+	$81$		$x^{4}$	-	$9 x^{3}$	+	$95 x^{2}$	-	$729 x$	+	$1134$
					-	$x^{4}$	-	$0$	-	$81 x^{2}$
							-	$9 x^{3}$	+	$14 x^{2}$	-	$729 x$
							-	$9 x^{3}$	+	$0$	-	$729 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{3} + 0 - 729 x$

$x^{2}$

$9 x$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

$9 x^{3}$

$0$

$729 x$

Step 10

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

$x^{2}$

$9 x$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

$9 x^{3}$

$0$

$729 x$

$14 x^{2}$

$0$

Step 11

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

$x^{2}$

$9 x$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

$9 x^{3}$

$0$

$729 x$

$14 x^{2}$

$0$

$1134$

Step 12

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

$x^{2}$

$9 x$

$14$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

$9 x^{3}$

$0$

$729 x$

$14 x^{2}$

$0$

$1134$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$9 x$

$14$

$x^{2}$

$0 x$

$81$

$x^{4}$

$9 x^{3}$

$95 x^{2}$

$729 x$

$1134$

$x^{4}$

$0$

$81 x^{2}$

$9 x^{3}$

$14 x^{2}$

$729 x$

$9 x^{3}$

$0$

$729 x$

$14 x^{2}$

$0$

$1134$

$14 x^{2}$

$0$

$1134$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{2} + 0 + 1134$

						$x^{2}$	-	$9 x$	+	$14$
$x^{2}$	+	$0 x$	+	$81$		$x^{4}$	-	$9 x^{3}$	+	$95 x^{2}$	-	$729 x$	+	$1134$
					-	$x^{4}$	-	$0$	-	$81 x^{2}$
							-	$9 x^{3}$	+	$14 x^{2}$	-	$729 x$
							+	$9 x^{3}$	-	$0$	+	$729 x$
									+	$14 x^{2}$	+	$0$	+	$1134$
									-	$14 x^{2}$	-	$0$	-	$1134$

Step 15

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

						$x^{2}$	-	$9 x$	+	$14$
$x^{2}$	+	$0 x$	+	$81$		$x^{4}$	-	$9 x^{3}$	+	$95 x^{2}$	-	$729 x$	+	$1134$
					-	$x^{4}$	-	$0$	-	$81 x^{2}$
							-	$9 x^{3}$	+	$14 x^{2}$	-	$729 x$
							+	$9 x^{3}$	-	$0$	+	$729 x$
									+	$14 x^{2}$	+	$0$	+	$1134$
									-	$14 x^{2}$	-	$0$	-	$1134$
														$0$

Step 16

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

$x^{2} - 9 x + 14$