Pre-Algebra Examples

$\frac{(x + 3) (x^{2} - 3 x) + x (x - 3)}{(x + 3) (x - 3)}$

Step 1

Expand $(x + 3) (x^{2} - 3 x) + x (x - 3)$ .

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Step 1.1

Apply the distributive property.

$\frac{x (x^{2} - 3 x) + 3 (x^{2} - 3 x) + x (x - 3)}{(x + 3) (x - 3)}$

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Remove parentheses.

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

Step 1.6

Reorder $x$ and $- 3$ .

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

Step 1.7

Remove parentheses.

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

Step 1.8

Reorder $x$ and $- 3$ .

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Add $1$ and $2$ .

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

Add $1$ and $1$ .

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

Step 1.16

Multiply $3$ by $- 3$ .

$\frac{x^{3} - 3 x^{2} + 3 x^{2} - 9 x + x \cdot x - 3 x}{(x + 3) (x - 3)}$

Step 1.17

Add $- 3 x^{2}$ and $3 x^{2}$ .

$\frac{x^{3} + 0 - 9 x + x \cdot x - 3 x}{(x + 3) (x - 3)}$

Step 1.18

Subtract $9 x$ from $0$ .

$\frac{x^{3} - 9 x + x \cdot x - 3 x}{(x + 3) (x - 3)}$

Step 1.19

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

$\frac{x^{3} - 9 x + x \cdot x - 3 x}{(x + 3) (x - 3)}$

Step 1.20

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

$\frac{x^{3} - 9 x + x \cdot x - 3 x}{(x + 3) (x - 3)}$

Step 1.21

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

$\frac{x^{3} - 9 x + x^{1 + 1} - 3 x}{(x + 3) (x - 3)}$

Step 1.22

Add $1$ and $1$ .

$\frac{x^{3} - 9 x + x^{2} - 3 x}{(x + 3) (x - 3)}$

Step 1.23

Move $- 9 x$ .

$\frac{x^{3} + x^{2} - 3 x - 9 x}{(x + 3) (x - 3)}$

Step 1.24

Subtract $9 x$ from $- 3 x$ .

$\frac{x^{3} + x^{2} - 12 x}{(x + 3) (x - 3)}$

Step 2

Expand $(x + 3) (x - 3)$ .

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Step 2.1

Apply the distributive property.

$\frac{x^{3} + x^{2} - 12 x}{x (x - 3) + 3 (x - 3)}$

Step 2.2

Apply the distributive property.

$\frac{x^{3} + x^{2} - 12 x}{x \cdot x + x \cdot - 3 + 3 (x - 3)}$

Step 2.3

Apply the distributive property.

$\frac{x^{3} + x^{2} - 12 x}{x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3}$

Step 2.4

Reorder $x$ and $- 3$ .

$\frac{x^{3} + x^{2} - 12 x}{x \cdot x - 3 x + 3 x + 3 \cdot - 3}$

Step 2.5

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

$\frac{x^{3} + x^{2} - 12 x}{x \cdot x - 3 x + 3 x + 3 \cdot - 3}$

Step 2.6

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

$\frac{x^{3} + x^{2} - 12 x}{x \cdot x - 3 x + 3 x + 3 \cdot - 3}$

Step 2.7

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

$\frac{x^{3} + x^{2} - 12 x}{x^{1 + 1} - 3 x + 3 x + 3 \cdot - 3}$

Step 2.8

Add $1$ and $1$ .

$\frac{x^{3} + x^{2} - 12 x}{x^{2} - 3 x + 3 x + 3 \cdot - 3}$

Step 2.9

Multiply $3$ by $- 3$ .

$\frac{x^{3} + x^{2} - 12 x}{x^{2} - 3 x + 3 x - 9}$

Step 2.10

Add $- 3 x$ and $3 x$ .

$\frac{x^{3} + x^{2} - 12 x}{x^{2} + 0 - 9}$

Step 2.11

Subtract $9$ from $0$ .

$\frac{x^{3} + x^{2} - 12 x}{x^{2} - 9}$

Step 3

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

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

Step 4

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

$x$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

Step 5

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

Step 6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - 9 x$

$x$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

Step 7

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

$x$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

$x^{2}$

$3 x$

Step 8

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

$x$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

$x^{2}$

$3 x$

$0$

Step 9

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

$x$

$1$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

$x^{2}$

$3 x$

$0$

Step 10

Multiply the new quotient term by the divisor.

$x$

$1$

$x^{2}$

$0 x$

$9$

$x^{3}$

$x^{2}$

$12 x$

$0$

$x^{3}$

$0$

$9 x$

$x^{2}$

$3 x$

$0$

$x^{2}$

$0$

$9$

Step 11

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 - 9$

						$x$	+	$1$
$x^{2}$	+	$0 x$	-	$9$		$x^{3}$	+	$x^{2}$	-	$12 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							+	$x^{2}$	-	$3 x$	+	$0$
							-	$x^{2}$	-	$0$	+	$9$

Step 12

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

						$x$	+	$1$
$x^{2}$	+	$0 x$	-	$9$		$x^{3}$	+	$x^{2}$	-	$12 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							+	$x^{2}$	-	$3 x$	+	$0$
							-	$x^{2}$	-	$0$	+	$9$
									-	$3 x$	+	$9$

Step 13

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

$x + 1 + \frac{- 3 x + 9}{x^{2} - 9}$