Calculus Examples

$\frac{x^{5} - x^{4} + x^{3} + 2 x^{2} - x + 4}{x^{3} + 1}$

Step 1

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

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

Step 2

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x^{3}$ .

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} + 0 + 0 + x^{2}$

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

Step 5

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

Step 6

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

Step 7

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

$x^{2}$

$x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

Step 9

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

$x^{2}$

$x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

Step 10

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

$x^{2}$

$x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

$x^{3}$

$x^{2}$

$0$

Step 11

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

$x^{2}$

$x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

$x^{3}$

$x^{2}$

$0$

$4$

Step 12

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

$x^{2}$

$x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

$x^{3}$

$x^{2}$

$0$

$4$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$x^{3}$

$2 x^{2}$

$x$

$4$

$x^{5}$

$0$

$x^{2}$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$x^{4}$

$0$

$x$

$x^{3}$

$x^{2}$

$0$

$4$

$x^{3}$

$0$

$1$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 + 0 + 1$

								$x^{2}$	-	$x$	+	$1$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	-	$x^{4}$	+	$x^{3}$	+	$2 x^{2}$	-	$x$	+	$4$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									-	$x^{4}$	+	$x^{3}$	+	$x^{2}$	-	$x$
									+	$x^{4}$	-	$0$	-	$0$	+	$x$
											+	$x^{3}$	+	$x^{2}$	+	$0$	+	$4$
											-	$x^{3}$	-	$0$	-	$0$	-	$1$

Step 15

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

								$x^{2}$	-	$x$	+	$1$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	-	$x^{4}$	+	$x^{3}$	+	$2 x^{2}$	-	$x$	+	$4$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									-	$x^{4}$	+	$x^{3}$	+	$x^{2}$	-	$x$
									+	$x^{4}$	-	$0$	-	$0$	+	$x$
											+	$x^{3}$	+	$x^{2}$	+	$0$	+	$4$
											-	$x^{3}$	-	$0$	-	$0$	-	$1$
													+	$x^{2}$	+	$0$	+	$3$

Step 16

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

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