Algebra Examples

$(x^{5} + 2 x^{4} - 6 x^{3} + x^{2} - 5 x + 1) \div (x^{3} + 1)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

Step 1.2

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

Step 1.4

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

Step 1.5

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

$2 x^{4}$

$6 x^{3}$

$0$

Step 1.6

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

$x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

$2 x^{4}$

$6 x^{3}$

$0$

$5 x$

$1$

Step 1.7

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $x^{3}$ .

$x^{2}$

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

$2 x^{4}$

$6 x^{3}$

$0$

$5 x$

$1$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$2 x^{4}$

$6 x^{3}$

$x^{2}$

$5 x$

$1$

$x^{5}$

$0$

$x^{2}$

$2 x^{4}$

$6 x^{3}$

$0$

$5 x$

$1$

$2 x^{4}$

$0$

$2 x$

Step 1.9

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

								$x^{2}$	+	$2 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$

Step 1.10

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

								$x^{2}$	+	$2 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$

Step 1.11

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

								$x^{2}$	+	$2 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$	+	$1$

Step 1.12

Divide the highest order term in the dividend $- 6 x^{3}$ by the highest order term in divisor $x^{3}$ .

								$x^{2}$	+	$2 x$	-	$6$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$	+	$1$

Step 1.13

Multiply the new quotient term by the divisor.

								$x^{2}$	+	$2 x$	-	$6$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$	+	$1$
											-	$6 x^{3}$	+	$0$	+	$0$	-	$6$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{3} + 0 + 0 - 6$

								$x^{2}$	+	$2 x$	-	$6$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$	+	$1$
											+	$6 x^{3}$	-	$0$	-	$0$	+	$6$

Step 1.15

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

								$x^{2}$	+	$2 x$	-	$6$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$1$		$x^{5}$	+	$2 x^{4}$	-	$6 x^{3}$	+	$x^{2}$	-	$5 x$	+	$1$
							-	$x^{5}$	-	$0$	-	$0$	-	$x^{2}$
									+	$2 x^{4}$	-	$6 x^{3}$	+	$0$	-	$5 x$	+	$1$
									-	$2 x^{4}$	-	$0$	-	$0$	-	$2 x$
											-	$6 x^{3}$	+	$0$	-	$7 x$	+	$1$
											+	$6 x^{3}$	-	$0$	-	$0$	+	$6$
															-	$7 x$	+	$7$

Step 1.16

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

$x^{2} + 2 x - 6 + \frac{- 7 x + 7}{x^{3} + 1}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$- 7 x + 7$