Pre-Algebra Examples

x^{3} - 5 x^{2} + 3 x - 15

$x^{3} - 5 x^{2} + 3 x - 15$ ,

x + 3

$x + 3$

Step 1

Divide the first expression by the second expression.

\frac{x^{3} - 5 x^{2} + 3 x - 15}{x + 3}

$\frac{x^{3} - 5 x^{2} + 3 x - 15}{x + 3}$

Step 2

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

$x$

3

$3$

x^{3}

$x^{3}$

5 x^{2}

$5 x^{2}$

3 x

$3 x$

15

$15$

Step 3

Divide the highest order term in the dividend

x^{3}

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

x

$x$ .

					$x^{2}$ $x^{2}$
	$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$

Step 4

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			+	$x^{3}$ $x^{3}$	+	$3 x^{2}$ $3 x^{2}$

Step 5

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

x^{3} + 3 x^{2}

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$

Step 6

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$

Step 7

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$

Step 8

Divide the highest order term in the dividend

- 8 x^{2}

$- 8 x^{2}$ by the highest order term in divisor

x

$x$ .

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$

Step 9

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					-	$8 x^{2}$ $8 x^{2}$	-	$24 x$ $24 x$

Step 10

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

- 8 x^{2} - 24 x

$- 8 x^{2} - 24 x$

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$

Step 11

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

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$

Step 12

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

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$	-	$15$ $15$

Step 13

Divide the highest order term in the dividend

27 x

$27 x$ by the highest order term in divisor

x

$x$ .

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$	+	$27$ $27$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$	-	$15$ $15$

Step 14

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$	+	$27$ $27$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$	-	$15$ $15$
							+	$27 x$ $27 x$	+	$81$ $81$

Step 15

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

27 x + 81

$27 x + 81$

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$	+	$27$ $27$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$	-	$15$ $15$
							-	$27 x$ $27 x$	-	$81$ $81$

Step 16

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

				$x^{2}$ $x^{2}$	-	$8 x$ $8 x$	+	$27$ $27$
$x$ $x$	+	$3$ $3$		$x^{3}$ $x^{3}$	-	$5 x^{2}$ $5 x^{2}$	+	$3 x$ $3 x$	-	$15$ $15$
			-	$x^{3}$ $x^{3}$	-	$3 x^{2}$ $3 x^{2}$
					-	$8 x^{2}$ $8 x^{2}$	+	$3 x$ $3 x$
					+	$8 x^{2}$ $8 x^{2}$	+	$24 x$ $24 x$
							+	$27 x$ $27 x$	-	$15$ $15$
							-	$27 x$ $27 x$	-	$81$ $81$
									-	$96$ $96$

Step 17

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

x^{2} - 8 x + 27 - \frac{96}{x + 3}

$x^{2} - 8 x + 27 - \frac{96}{x + 3}$

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