Pre-Algebra Examples

$\frac{(9 - x) (4 + x)}{(x - 9) (x + 1)}$

Step 1

Expand $(9 - x) (4 + x)$ .

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Step 1.1

Apply the distributive property.

$\frac{9 (4 + x) - x (4 + x)}{(x - 9) (x + 1)}$

Step 1.2

Apply the distributive property.

$\frac{9 \cdot 4 + 9 x - x (4 + x)}{(x - 9) (x + 1)}$

Step 1.3

Apply the distributive property.

$\frac{9 \cdot 4 + 9 x - x \cdot 4 - x \cdot x}{(x - 9) (x + 1)}$

Step 1.4

Move $x$ .

$\frac{9 \cdot 4 + 9 x - 1 \cdot (4 x) - x \cdot x}{(x - 9) (x + 1)}$

Step 1.5

Multiply $9$ by $4$ .

$\frac{36 + 9 x - 1 \cdot (4 x) - x \cdot x}{(x - 9) (x + 1)}$

Step 1.6

Multiply $- 1$ by $4$ .

$\frac{36 + 9 x - 4 x - x \cdot x}{(x - 9) (x + 1)}$

Step 1.7

Factor out negative.

$\frac{36 + 9 x - 4 x - (x \cdot x)}{(x - 9) (x + 1)}$

Step 1.8

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

$\frac{36 + 9 x - 4 x - (x \cdot x)}{(x - 9) (x + 1)}$

Step 1.9

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

$\frac{36 + 9 x - 4 x - (x \cdot x)}{(x - 9) (x + 1)}$

Step 1.10

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

$\frac{36 + 9 x - 4 x - x^{1 + 1}}{(x - 9) (x + 1)}$

Step 1.11

Add $1$ and $1$ .

$\frac{36 + 9 x - 4 x - x^{2}}{(x - 9) (x + 1)}$

Step 1.12

Subtract $4 x$ from $9 x$ .

$\frac{36 + 5 x - x^{2}}{(x - 9) (x + 1)}$

Step 1.13

Reorder $5 x$ and $- x^{2}$ .

$\frac{36 - x^{2} + 5 x}{(x - 9) (x + 1)}$

Step 1.14

Move $36$ .

$\frac{- x^{2} + 5 x + 36}{(x - 9) (x + 1)}$

Step 2

Expand $(x - 9) (x + 1)$ .

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Step 2.1

Apply the distributive property.

$\frac{- x^{2} + 5 x + 36}{x (x + 1) - 9 (x + 1)}$

Step 2.2

Apply the distributive property.

$\frac{- x^{2} + 5 x + 36}{x \cdot x + x \cdot 1 - 9 (x + 1)}$

Step 2.3

Apply the distributive property.

$\frac{- x^{2} + 5 x + 36}{x \cdot x + x \cdot 1 - 9 x - 9 \cdot 1}$

Step 2.4

Reorder $x$ and $1$ .

$\frac{- x^{2} + 5 x + 36}{x \cdot x + 1 x - 9 x - 9 \cdot 1}$

Step 2.5

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

$\frac{- x^{2} + 5 x + 36}{x \cdot x + 1 x - 9 x - 9 \cdot 1}$

Step 2.6

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

$\frac{- x^{2} + 5 x + 36}{x \cdot x + 1 x - 9 x - 9 \cdot 1}$

Step 2.7

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

$\frac{- x^{2} + 5 x + 36}{x^{1 + 1} + 1 x - 9 x - 9 \cdot 1}$

Step 2.8

Add $1$ and $1$ .

$\frac{- x^{2} + 5 x + 36}{x^{2} + 1 x - 9 x - 9 \cdot 1}$

Step 2.9

Multiply $x$ by $1$ .

$\frac{- x^{2} + 5 x + 36}{x^{2} + x - 9 x - 9 \cdot 1}$

Step 2.10

Multiply $- 9$ by $1$ .

$\frac{- x^{2} + 5 x + 36}{x^{2} + x - 9 x - 9}$

Step 2.11

Subtract $9 x$ from $x$ .

$\frac{- x^{2} + 5 x + 36}{x^{2} - 8 x - 9}$

Step 3

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

$x^{2}$

$8 x$

$9$

$x^{2}$

$5 x$

$36$

Step 4

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

						-	$1$
	$x^{2}$	-	$8 x$	-	$9$	-	$x^{2}$	+	$5 x$	+	$36$

Step 5

Multiply the new quotient term by the divisor.

					-	$1$
$x^{2}$	-	$8 x$	-	$9$	-	$x^{2}$	+	$5 x$	+	$36$
					-	$x^{2}$	+	$8 x$	+	$9$

Step 6

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} + 8 x + 9$

					-	$1$
$x^{2}$	-	$8 x$	-	$9$	-	$x^{2}$	+	$5 x$	+	$36$
					+	$x^{2}$	-	$8 x$	-	$9$

Step 7

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

					-	$1$
$x^{2}$	-	$8 x$	-	$9$	-	$x^{2}$	+	$5 x$	+	$36$
					+	$x^{2}$	-	$8 x$	-	$9$
							-	$3 x$	+	$27$

Step 8

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

$- 1 + \frac{- 3 x + 27}{x^{2} - 8 x - 9}$