Algebra Examples

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

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

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

$0$ .

x

$x$

2

$2$

4 x^{3}

$4 x^{3}$

3 x^{2}

$3 x^{2}$

x

$x$

1

$1$

Step 2

Divide the highest order term in the dividend

4 x^{3}

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

x

$x$ .

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

Step 3

Multiply the new quotient term by the divisor.

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

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

4 x^{3} + 8 x^{2}

$4 x^{3} + 8 x^{2}$

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

Step 5

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

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$

Step 6

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

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$

Step 7

Divide the highest order term in the dividend

- 11 x^{2}

$- 11 x^{2}$ by the highest order term in divisor

x

$x$ .

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					-	$11 x^{2}$ $11 x^{2}$	-	$22 x$ $22 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

- 11 x^{2} - 22 x

$- 11 x^{2} - 22 x$

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$

Step 10

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

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$

Step 11

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

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$	+	$1$ $1$

Step 12

Divide the highest order term in the dividend

23 x

$23 x$ by the highest order term in divisor

x

$x$ .

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$23$ $23$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$	+	$1$ $1$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$23$ $23$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$	+	$1$ $1$
							+	$23 x$ $23 x$	+	$46$ $46$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

23 x + 46

$23 x + 46$

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$23$ $23$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$	+	$1$ $1$
							-	$23 x$ $23 x$	-	$46$ $46$

Step 15

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

				$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$23$ $23$
$x$ $x$	+	$2$ $2$		$4 x^{3}$ $4 x^{3}$	-	$3 x^{2}$ $3 x^{2}$	+	$x$ $x$	+	$1$ $1$
			-	$4 x^{3}$ $4 x^{3}$	-	$8 x^{2}$ $8 x^{2}$
					-	$11 x^{2}$ $11 x^{2}$	+	$x$ $x$
					+	$11 x^{2}$ $11 x^{2}$	+	$22 x$ $22 x$
							+	$23 x$ $23 x$	+	$1$ $1$
							-	$23 x$ $23 x$	-	$46$ $46$
									-	$45$ $45$

Step 16

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

4 x^{2} - 11 x + 23 - \frac{45}{x + 2}

$4 x^{2} - 11 x + 23 - \frac{45}{x + 2}$