Finite Math Examples

$\frac{4 x^{3} + 21 x^{2} - 42 x + 49}{x + 7}$

Step 1

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

$x$

$7$

$4 x^{3}$

$21 x^{2}$

$42 x$

$49$

Step 2

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

					$4 x^{2}$
	$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$

Step 3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			+	$4 x^{3}$	+	$28 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 28 x^{2}$

				$4 x^{2}$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$

Step 5

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

				$4 x^{2}$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$

Step 6

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

				$4 x^{2}$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$

Step 7

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

				$4 x^{2}$	-	$7 x$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$7 x$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					-	$7 x^{2}$	-	$49 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} - 49 x$

				$4 x^{2}$	-	$7 x$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$

Step 10

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

				$4 x^{2}$	-	$7 x$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$

Step 11

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

				$4 x^{2}$	-	$7 x$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$	+	$49$

Step 12

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$7 x$	+	$7$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$	+	$49$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$7 x$	+	$7$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$	+	$49$
							+	$7 x$	+	$49$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 49$

				$4 x^{2}$	-	$7 x$	+	$7$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$	+	$49$
							-	$7 x$	-	$49$

Step 15

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

				$4 x^{2}$	-	$7 x$	+	$7$
$x$	+	$7$		$4 x^{3}$	+	$21 x^{2}$	-	$42 x$	+	$49$
			-	$4 x^{3}$	-	$28 x^{2}$
					-	$7 x^{2}$	-	$42 x$
					+	$7 x^{2}$	+	$49 x$
							+	$7 x$	+	$49$
							-	$7 x$	-	$49$
										$0$

Step 16

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

$4 x^{2} - 7 x + 7$