Algebra Examples

$h (x) = - \frac{44 x^{2}}{v^{2}} + x + 6$

Step 1

Find where the expression $- \frac{44 x^{2}}{v^{2}} + x + 6$ is undefined.

$x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

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

Step 4

Find $n$ and $m$ .

$n = 2$

$m = 0$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Combine.

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Step 6.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{v^{2}}{v^{2}}$ .

$- \frac{44 x^{2}}{v^{2}} + \frac{x v^{2}}{v^{2}} + 6$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{- 44 x^{2} + x v^{2}}{v^{2}} + 6$

Step 6.1.3

Factor $x$ out of $- 44 x^{2} + x v^{2}$ .

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Step 6.1.3.1

Factor $x$ out of $- 44 x^{2}$ .

$\frac{x (- 44 x) + x v^{2}}{v^{2}} + 6$

Step 6.1.3.2

Factor $x$ out of $x v^{2}$ .

$\frac{x (- 44 x) + x (v^{2})}{v^{2}} + 6$

Step 6.1.3.3

Factor $x$ out of $x (- 44 x) + x (v^{2})$ .

$\frac{x (- 44 x + v^{2})}{v^{2}} + 6$

Step 6.1.4

To write $6$ as a fraction with a common denominator, multiply by $\frac{v^{2}}{v^{2}}$ .

$\frac{x (- 44 x + v^{2})}{v^{2}} + \frac{6 v^{2}}{v^{2}}$

Step 6.1.5

Combine the numerators over the common denominator.

$\frac{x (- 44 x + v^{2}) + 6 v^{2}}{v^{2}}$

Step 6.1.6

Simplify the numerator.

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Step 6.1.6.1

Apply the distributive property.

$\frac{x (- 44 x) + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.1.6.2

Rewrite using the commutative property of multiplication.

$\frac{- 44 x \cdot x + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.1.6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.6.3.1

Move $x$ .

$\frac{- 44 (x \cdot x) + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.1.6.3.2

Multiply $x$ by $x$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.1.7

Simplify with factoring out.

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Step 6.1.7.1

Factor $- 1$ out of $- 44 x^{2}$ .

$\frac{- (44 x^{2}) + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.1.7.2

Factor $- 1$ out of $x v^{2}$ .

$\frac{- (44 x^{2}) - (- x v^{2}) + 6 v^{2}}{v^{2}}$

Step 6.1.7.3

Factor $- 1$ out of $- (44 x^{2}) - (- x v^{2})$ .

$\frac{- (44 x^{2} - x v^{2}) + 6 v^{2}}{v^{2}}$

Step 6.1.7.4

Factor $- 1$ out of $6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Step 6.1.7.5

Factor $- 1$ out of $- (44 x^{2} - x v^{2}) - (- 6 v^{2})$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Step 6.1.7.6

Simplify the expression.

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Step 6.1.7.6.1

Rewrite $- (44 x^{2} - x v^{2} - 6 v^{2})$ as $- 1 (44 x^{2} - x v^{2} - 6 v^{2})$ .

$\frac{- 1 (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Step 6.1.7.6.2

Move the negative in front of the fraction.

$- \frac{44 x^{2} - x v^{2} - 6 v^{2}}{v^{2}}$

Step 6.1.8

Simplify.

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Factor $- 1$ out of $- 44 x^{2}$ .

$\frac{- (44 x^{2}) + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.2.2

Factor $- 1$ out of $x v^{2}$ .

$\frac{- (44 x^{2}) - (- x v^{2}) + 6 v^{2}}{v^{2}}$

Step 6.2.3

Factor $- 1$ out of $- (44 x^{2}) - (- x v^{2})$ .

$\frac{- (44 x^{2} - x v^{2}) + 6 v^{2}}{v^{2}}$

Step 6.2.4

Factor $- 1$ out of $6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Step 6.2.5

Factor $- 1$ out of $- (44 x^{2} - x v^{2}) - (- 6 v^{2})$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Step 6.2.6

Simplify the expression.

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Step 6.2.6.1

Rewrite $- (44 x^{2} - x v^{2} - 6 v^{2})$ as $- 1 (44 x^{2} - x v^{2} - 6 v^{2})$ .

$\frac{- 1 (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Step 6.2.6.2

Move the negative in front of the fraction.

$- \frac{44 x^{2} - x v^{2} - 6 v^{2}}{v^{2}}$

Step 6.3

Expand $- (44 x^{2} - x v^{2} - 6 v^{2})$ .

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Step 6.3.1

Negate $44 x^{2} - x v^{2} - 6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Step 6.3.2

Apply the distributive property.

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Step 6.3.3

Apply the distributive property.

$\frac{- (44 x^{2}) - (- x v^{2}) - (- 6 v^{2})}{v^{2}}$

Step 6.3.4

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) - (- x v^{2}) - (- 6 v^{2})}{v^{2}}$

Step 6.3.5

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) + x v^{2} - (- 6 v^{2})}{v^{2}}$

Step 6.3.6

Move parentheses.

$\frac{- 1 \cdot (44 x^{2}) + 1 x v^{2} - (- 6 v^{2})}{v^{2}}$

Step 6.3.7

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.3.8

Multiply $- 1$ by $44$ .

$\frac{- 44 x^{2} + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.3.9

Multiply $- 1$ by $- 1$ .

$\frac{- 44 x^{2} + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.3.10

Multiply $x$ by $1$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.3.11

Multiply $- 1$ by $- 6$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Step 6.3.12

Move $x v^{2}$ .

$\frac{- 44 x^{2} + 6 v^{2} + x v^{2}}{v^{2}}$

Step 6.3.13

Reorder $- 44 x^{2}$ and $6 v^{2}$ .

$\frac{6 v^{2} - 44 x^{2} + x v^{2}}{v^{2}}$

Step 6.4

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

$v^{2}$

$44 x^{2}$

$v^{2} x$

$6 v^{2}$

Step 6.5

Divide the highest order term in the dividend $- 44 x^{2}$ by the highest order term in divisor $v^{2}$ .

		-	$\frac{44 x^{2}}{v^{2}}$
	$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$

Step 6.6

Multiply the new quotient term by the divisor.

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	-	$44 x^{2}$

Step 6.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 44 x^{2}$

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$

Step 6.8

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

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$

Step 6.9

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

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$

Step 6.10

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$

Step 6.11

Multiply the new quotient term by the divisor.

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			+	$x v^{2}$

Step 6.12

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$

Step 6.13

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$

Step 6.14

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$

Step 6.15

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

$- \frac{44 x^{2}}{v^{2}} + x + \frac{6 v^{2}}{v^{2}}$

Step 6.16

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

$y = - \frac{44 x^{2}}{v^{2}} + x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{44 x^{2}}{v^{2}} + x$

Step 8