Finite Math Examples

$\frac{4 x^{3} - 7 x^{2} - 11 x + 5}{4 x + 5}$

Step 1

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

$4 x$

$5$

$4 x^{3}$

$7 x^{2}$

$11 x$

$5$

Step 2

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

					$x^{2}$
	$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			+	$4 x^{3}$	+	$5 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 5 x^{2}$

				$x^{2}$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$

Step 5

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

				$x^{2}$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$

Step 6

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

				$x^{2}$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$

Step 7

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

				$x^{2}$	-	$3 x$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					-	$12 x^{2}$	-	$15 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} - 15 x$

				$x^{2}$	-	$3 x$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$

Step 10

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

				$x^{2}$	-	$3 x$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$

Step 11

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

				$x^{2}$	-	$3 x$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$	+	$5$

Step 12

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

				$x^{2}$	-	$3 x$	+	$1$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$	+	$5$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	+	$1$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$	+	$5$
							+	$4 x$	+	$5$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 5$

				$x^{2}$	-	$3 x$	+	$1$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$	+	$5$
							-	$4 x$	-	$5$

Step 15

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

				$x^{2}$	-	$3 x$	+	$1$
$4 x$	+	$5$		$4 x^{3}$	-	$7 x^{2}$	-	$11 x$	+	$5$
			-	$4 x^{3}$	-	$5 x^{2}$
					-	$12 x^{2}$	-	$11 x$
					+	$12 x^{2}$	+	$15 x$
							+	$4 x$	+	$5$
							-	$4 x$	-	$5$
										$0$

Step 16

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

$x^{2} - 3 x + 1$