Algebra Examples

$(x - 1) (x^{2} + x + 1) \div (x + 2)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

Expand $(x - 1) (x^{2} + x + 1)$ .

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Step 1.1.1

Apply the distributive property.

$\frac{x (x^{2} + x + 1) - 1 (x^{2} + x + 1)}{x + 2}$

Step 1.1.2

Apply the distributive property.

$\frac{x (x^{2} + x) + x \cdot 1 - 1 (x^{2} + x + 1)}{x + 2}$

Step 1.1.3

Apply the distributive property.

$\frac{x \cdot x^{2} + x \cdot x + x \cdot 1 - 1 (x^{2} + x + 1)}{x + 2}$

Step 1.1.4

Apply the distributive property.

$\frac{x \cdot x^{2} + x \cdot x + x \cdot 1 - 1 (x^{2} + x) - 1 \cdot 1}{x + 2}$

Step 1.1.5

Apply the distributive property.

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

Step 1.1.6

Reorder $x$ and $1$ .

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

Step 1.1.7

Raise $x$ to the power of $1$ .

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

Step 1.1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.9

Add $1$ and $2$ .

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

Step 1.1.10

Raise $x$ to the power of $1$ .

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

Step 1.1.11

Raise $x$ to the power of $1$ .

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

Step 1.1.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.13

Add $1$ and $1$ .

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

Step 1.1.14

Multiply $x$ by $1$ .

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

Step 1.1.15

Multiply $- 1$ by $1$ .

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

Step 1.1.16

Move $x$ .

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

Step 1.1.17

Subtract $1 x^{2}$ from $x^{2}$ .

$\frac{x^{3} + 0 + x - 1 x - 1}{x + 2}$

Step 1.1.18

Add $x^{3}$ and $0$ .

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

Step 1.1.19

Subtract $1 x$ from $x$ .

$\frac{x^{3} + 0 - 1}{x + 2}$

Step 1.1.20

Add $x^{3}$ and $0$ .

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

Step 1.2

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

$x$

$2$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.3

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

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

Step 1.4

Multiply the new quotient term by the divisor.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Multiply the new quotient term by the divisor.

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	+	$4$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$4 x$
							+	$4 x$	-	$1$
							+	$4 x$	+	$8$

Step 1.15

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 8$

				$x^{2}$	-	$2 x$	+	$4$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$4 x$
							+	$4 x$	-	$1$
							-	$4 x$	-	$8$

Step 1.16

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

				$x^{2}$	-	$2 x$	+	$4$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$4 x$
							+	$4 x$	-	$1$
							-	$4 x$	-	$8$
									-	$9$

Step 1.17

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

$x^{2} - 2 x + 4 - \frac{9}{x + 2}$

Step 2

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

$- 9$