Pre-Algebra Examples

\frac{\frac{x^{2} - 25}{x^{2} - 16}}{\frac{2 x + 10}{x^{2} - 4 x \cdot \frac{2 x + 8}{x^{2} - 5 x}}}

$\frac{\frac{x^{2} - 25}{x^{2} - 16}}{\frac{2 x + 10}{x^{2} - 4 x \cdot \frac{2 x + 8}{x^{2} - 5 x}}}$

Step 1

Find where the expression

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$ is undefined.

x = - 4, x = 4

$x = - 4, x = 4$

Step 2

Since

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

\to

$\to$

- \infty

$- \infty$ as

x

$x$

\to

$\to$

- 4

$- 4$ from the left and

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

- 4

$- 4$ from the right, then

x = - 4

$x = - 4$ is a vertical asymptote.

x = - 4

$x = - 4$

Step 3

Since

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

4

$4$ from the left and

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

\to

$\to$

- \infty

$- \infty$ as

x

$x$

\to

$\to$

4

$4$ from the right, then

x = 4

$x = 4$ is a vertical asymptote.

x = 4

$x = 4$

Step 4

List all of the vertical asymptotes:

x = - 4, 4

$x = - 4, 4$

Step 5

Consider the rational function

R (x) = \frac{a x^{n}}{b x^{m}}

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

$n < m$ , then the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

$n = m$ , then the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ .

3. If

n > m

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 6

Find

n

$n$ and

m

$m$ .

n = 3

$n = 3$

m = 2

$m = 2$

Step 7

Since

n > m

$n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Factor

2

$2$ out of

2 x^{2} - 32

$2 x^{2} - 32$ .

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Step 8.1.1.1

Factor

2

$2$ out of

2 x^{2}

$2 x^{2}$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2}) - 32}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2}) - 32}$

Step 8.1.1.2

Factor

2

$2$ out of

- 32

$- 32$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} + 2 \cdot - 16}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} + 2 \cdot - 16}$

Step 8.1.1.3

Factor

2

$2$ out of

2 x^{2} + 2 \cdot - 16

$2 x^{2} + 2 \cdot - 16$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 16)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 16)}$

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 16)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 16)}$

Step 8.1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 4^{2})}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x^{2} - 4^{2})}$

Step 8.1.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 4

$b = 4$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x + 4) (x - 4)}$

Step 8.2

Expand

2 (x + 4) (x - 4)

$2 (x + 4) (x - 4)$ .

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Step 8.2.1

Apply the distributive property.

\frac{x^{3} - 5 x^{2} - 8 x - 32}{(2 x + 2 \cdot 4) (x - 4)}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{(2 x + 2 \cdot 4) (x - 4)}$

Step 8.2.2

Apply the distributive property.

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x (x - 4) + 2 \cdot (4 (x - 4))}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x (x - 4) + 2 \cdot (4 (x - 4))}$

Step 8.2.3

Apply the distributive property.

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 x \cdot - 4 + 2 \cdot (4 (x - 4))}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 x \cdot - 4 + 2 \cdot (4 (x - 4))}$

Step 8.2.4

Apply the distributive property.

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 x \cdot - 4 + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 x \cdot - 4 + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.5

Move

x

$x$ .

\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x \cdot x + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.6

Raise

$x$ to the power of

$1$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x \cdot x) + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.7

Raise

$x$ to the power of

$1$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 (x \cdot x) + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.8

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{1 + 1} + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.9

Add

$1$ and

$1$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} + 2 \cdot (- 4 x) + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.10

Multiply

$2$ by

$- 4$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} - 8 x + 2 \cdot (4 x) + 2 \cdot 4 \cdot - 4}$

Step 8.2.11

Multiply

$2$ by

$4$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} - 8 x + 8 x + 2 \cdot 4 \cdot - 4}$

Step 8.2.12

Multiply

$2$ by

$4$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} - 8 x + 8 x + 8 \cdot - 4}$

Step 8.2.13

Multiply

$8$ by

$- 4$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} - 8 x + 8 x - 32}$

Step 8.2.14

Add

$- 8 x$ and

$8 x$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} + 0 - 32}$

Step 8.2.15

Subtract

$32$ from

$0$ .

$\frac{x^{3} - 5 x^{2} - 8 x - 32}{2 x^{2} - 32}$

Step 8.3

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

$0$ .

$2 x^{2}$

$0 x$

$32$

$x^{3}$

$5 x^{2}$

$8 x$

$32$

Step 8.4

Divide the highest order term in the dividend

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

$2 x^{2}$ .

							$\frac{x}{2}$
	$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$

Step 8.5

Multiply the new quotient term by the divisor.

						$\frac{x}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					+	$x^{3}$	+	$0$	-	$16 x$

Step 8.6

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

$x^{3} + 0 - 16 x$

						$\frac{x}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$

Step 8.7

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

						$\frac{x}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$

Step 8.8

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

						$\frac{x}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$	-	$32$

Step 8.9

Divide the highest order term in the dividend

$- 5 x^{2}$ by the highest order term in divisor

$2 x^{2}$ .

						$\frac{x}{2}$	-	$\frac{5}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$	-	$32$

Step 8.10

Multiply the new quotient term by the divisor.

						$\frac{x}{2}$	-	$\frac{5}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$	-	$32$
							-	$5 x^{2}$	+	$0$	+	$80$

Step 8.11

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

$- 5 x^{2} + 0 + 80$

						$\frac{x}{2}$	-	$\frac{5}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$	-	$32$
							+	$5 x^{2}$	-	$0$	-	$80$

Step 8.12

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

						$\frac{x}{2}$	-	$\frac{5}{2}$
$2 x^{2}$	+	$0 x$	-	$32$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	-	$32$
					-	$x^{3}$	-	$0$	+	$16 x$
							-	$5 x^{2}$	+	$8 x$	-	$32$
							+	$5 x^{2}$	-	$0$	-	$80$
									+	$8 x$	-	$112$

Step 8.13

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

$\frac{x}{2} - \frac{5}{2} + \frac{8 x - 112}{2 x^{2} - 32}$

Step 8.14

The oblique asymptote is the polynomial portion of the long division result.

$y = \frac{x}{2} - \frac{5}{2}$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes:

$x = - 4, 4$

No Horizontal Asymptotes

Oblique Asymptotes:

$y = \frac{x}{2} - \frac{5}{2}$

Step 10