Finite Math Examples

$\frac{x^{3} - 5 x^{2} + 2 x - 7}{x + 2}$

Step 1

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

$x$

$2$

$x^{3}$

$5 x^{2}$

$2 x$

$7$

Step 2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			+	$x^{3}$	+	$2 x^{2}$

Step 4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$

Step 5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$

Step 6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$

Step 7

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					-	$7 x^{2}$	-	$14 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} - 14 x$

				$x^{2}$	-	$7 x$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$

Step 10

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

				$x^{2}$	-	$7 x$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$

Step 11

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

				$x^{2}$	-	$7 x$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$	-	$7$

Step 12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$	-	$7$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$	-	$7$
							+	$16 x$	+	$32$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 32$

				$x^{2}$	-	$7 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$	-	$7$
							-	$16 x$	-	$32$

Step 15

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

				$x^{2}$	-	$7 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$5 x^{2}$	+	$2 x$	-	$7$
			-	$x^{3}$	-	$2 x^{2}$
					-	$7 x^{2}$	+	$2 x$
					+	$7 x^{2}$	+	$14 x$
							+	$16 x$	-	$7$
							-	$16 x$	-	$32$
									-	$39$

Step 16

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

$x^{2} - 7 x + 16 - \frac{39}{x + 2}$