Algebra Examples

$x^{2} + (y - {(\frac{4}{x})}^{2}) \cdot 2 = 1$

Step 1

Simplify.

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$2 y - \frac{32}{x^{2}} = 1 - x^{2}$

Step 1.1.2

Add $\frac{32}{x^{2}}$ to both sides of the equation.

$2 y = 1 - x^{2} + \frac{32}{x^{2}}$

Step 1.2

Divide each term in $2 y = 1 - x^{2} + \frac{32}{x^{2}}$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 y = 1 - x^{2} + \frac{32}{x^{2}}$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2} + \frac{- x^{2}}{2} + \frac{\frac{32}{x^{2}}}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2} + \frac{- x^{2}}{2} + \frac{\frac{32}{x^{2}}}{2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2} + \frac{- x^{2}}{2} + \frac{\frac{32}{x^{2}}}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{\frac{32}{x^{2}}}{2}$

Step 1.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{32}{x^{2}} \cdot \frac{1}{2}$

Step 1.2.3.1.3

Combine.

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{32 \cdot 1}{x^{2} \cdot 2}$

Step 1.2.3.1.4

Cancel the common factor of $32$ and $2$ .

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Step 1.2.3.1.4.1

Factor $2$ out of $32 \cdot 1$ .

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{2 (16 \cdot 1)}{x^{2} \cdot 2}$

Step 1.2.3.1.4.2

Cancel the common factors.

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Step 1.2.3.1.4.2.1

Factor $2$ out of $x^{2} \cdot 2$ .

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{2 (16 \cdot 1)}{2 \cdot x^{2}}$

Step 1.2.3.1.4.2.2

Cancel the common factor.

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{2 (16 \cdot 1)}{2 \cdot x^{2}}$

Step 1.2.3.1.4.2.3

Rewrite the expression.

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{16 \cdot 1}{x^{2}}$

Step 1.2.3.1.5

Multiply $16$ by $1$ .

$y = \frac{1}{2} - \frac{x^{2}}{2} + \frac{16}{x^{2}}$

Step 2

Find where the expression $\frac{1}{2} - \frac{x^{2}}{2} + \frac{16}{x^{2}}$ is undefined.

$x = 0$

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 $m$ .

$n = 4$

$m = 2$

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

Combine the numerators over the common denominator.

$\frac{1 - x^{2}}{2} + \frac{16}{x^{2}}$

Step 6.1.2

Simplify the numerator.

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Step 6.1.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - x^{2}}{2} + \frac{16}{x^{2}}$

Step 6.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{(1 + x) (1 - x)}{2} + \frac{16}{x^{2}}$

Step 6.1.3

To write $\frac{(1 + x) (1 - x)}{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{(1 + x) (1 - x)}{2} \cdot \frac{x^{2}}{x^{2}} + \frac{16}{x^{2}}$

Step 6.1.4

To write $\frac{16}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(1 + x) (1 - x)}{2} \cdot \frac{x^{2}}{x^{2}} + \frac{16}{x^{2}} \cdot \frac{2}{2}$

Step 6.1.5

Write each expression with a common denominator of $2 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.5.1

Multiply $\frac{(1 + x) (1 - x)}{2}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{(1 + x) (1 - x) x^{2}}{2 x^{2}} + \frac{16}{x^{2}} \cdot \frac{2}{2}$

Step 6.1.5.2

Multiply $\frac{16}{x^{2}}$ by $\frac{2}{2}$ .

$\frac{(1 + x) (1 - x) x^{2}}{2 x^{2}} + \frac{16 \cdot 2}{x^{2} \cdot 2}$

Step 6.1.5.3

Reorder the factors of $x^{2} \cdot 2$ .

$\frac{(1 + x) (1 - x) x^{2}}{2 x^{2}} + \frac{16 \cdot 2}{2 x^{2}}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{(1 + x) (1 - x) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7

Simplify the numerator.

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Step 6.1.7.1

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 6.1.7.1.1

Apply the distributive property.

$\frac{(1 (1 - x) + x (1 - x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.1.2

Apply the distributive property.

$\frac{(1 \cdot 1 + 1 (- x) + x (1 - x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.1.3

Apply the distributive property.

$\frac{(1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2

Simplify and combine like terms.

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Step 6.1.7.2.1

Simplify each term.

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Step 6.1.7.2.1.1

Multiply $1$ by $1$ .

$\frac{(1 + 1 (- x) + x \cdot 1 + x (- x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.1.2

Multiply $- x$ by $1$ .

$\frac{(1 - x + x \cdot 1 + x (- x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.1.3

Multiply $x$ by $1$ .

$\frac{(1 - x + x + x (- x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{(1 - x + x - x \cdot x) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.1.5

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

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Step 6.1.7.2.1.5.1

Move $x$ .

$\frac{(1 - x + x - (x \cdot x)) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.1.5.2

Multiply $x$ by $x$ .

$\frac{(1 - x + x - x^{2}) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.2

Add $- x$ and $x$ .

$\frac{(1 + 0 - x^{2}) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.2.3

Add $1$ and $0$ .

$\frac{(1 - x^{2}) x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.3

Apply the distributive property.

$\frac{1 x^{2} - x^{2} x^{2} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.4

Multiply $x^{2}$ by $1$ .

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

Step 6.1.7.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 6.1.7.5.1

Move $x^{2}$ .

$\frac{x^{2} - (x^{2} x^{2}) + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.5.2

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

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

Step 6.1.7.5.3

Add $2$ and $2$ .

$\frac{x^{2} - x^{4} + 16 \cdot 2}{2 x^{2}}$

Step 6.1.7.6

Multiply $16$ by $2$ .

$\frac{x^{2} - x^{4} + 32}{2 x^{2}}$

Step 6.1.7.7

Reorder terms.

$\frac{- x^{4} + x^{2} + 32}{2 x^{2}}$

Step 6.1.8

Simplify with factoring out.

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Step 6.1.8.1

Factor $- 1$ out of $- x^{4}$ .

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

Step 6.1.8.2

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

$\frac{- (x^{4}) - 1 (- x^{2}) + 32}{2 x^{2}}$

Step 6.1.8.3

Factor $- 1$ out of $- (x^{4}) - 1 (- x^{2})$ .

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

Step 6.1.8.4

Rewrite $32$ as $- 1 (- 32)$ .

$\frac{- (x^{4} - x^{2}) - 1 (- 32)}{2 x^{2}}$

Step 6.1.8.5

Factor $- 1$ out of $- (x^{4} - x^{2}) - 1 (- 32)$ .

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

Step 6.1.8.6

Simplify the expression.

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Step 6.1.8.6.1

Rewrite $- (x^{4} - x^{2} - 32)$ as $- 1 (x^{4} - x^{2} - 32)$ .

$\frac{- 1 (x^{4} - x^{2} - 32)}{2 x^{2}}$

Step 6.1.8.6.2

Move the negative in front of the fraction.

$- \frac{x^{4} - x^{2} - 32}{2 x^{2}}$

Step 6.1.9

Simplify.

$\frac{- x^{4} + x^{2} + 32}{2 x^{2}}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Factor $- 1$ out of $- x^{4}$ .

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

Step 6.2.2

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

$\frac{- (x^{4}) - 1 (- x^{2}) + 32}{2 x^{2}}$

Step 6.2.3

Factor $- 1$ out of $- (x^{4}) - 1 (- x^{2})$ .

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

Step 6.2.4

Rewrite $32$ as $- 1 (- 32)$ .

$\frac{- (x^{4} - x^{2}) - 1 (- 32)}{2 x^{2}}$

Step 6.2.5

Factor $- 1$ out of $- (x^{4} - x^{2}) - 1 (- 32)$ .

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

Step 6.2.6

Simplify the expression.

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Step 6.2.6.1

Rewrite $- (x^{4} - x^{2} - 32)$ as $- 1 (x^{4} - x^{2} - 32)$ .

$\frac{- 1 (x^{4} - x^{2} - 32)}{2 x^{2}}$

Step 6.2.6.2

Move the negative in front of the fraction.

$- \frac{x^{4} - x^{2} - 32}{2 x^{2}}$

Step 6.3

Expand $- (x^{4} - x^{2} - 32)$ .

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Step 6.3.1

Negate $x^{4} - x^{2} - 32$ .

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

Step 6.3.2

Apply the distributive property.

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

Step 6.3.3

Apply the distributive property.

$\frac{- x^{4} + x^{2} + 32}{2 x^{2}}$

Step 6.3.4

Move parentheses.

$\frac{- x^{4} + 1 x^{2} + 32}{2 x^{2}}$

Step 6.3.5

Multiply $- 1$ by $- 1$ .

$\frac{- x^{4} + 1 x^{2} + 32}{2 x^{2}}$

Step 6.3.6

Multiply $x^{2}$ by $1$ .

$\frac{- x^{4} + x^{2} + 32}{2 x^{2}}$

Step 6.3.7

Multiply $- 1$ by $- 32$ .

$\frac{- x^{4} + x^{2} + 32}{2 x^{2}}$

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 $0$ .

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

Step 6.5

Divide the highest order term in the dividend $- x^{4}$ by the highest order term in divisor $2 x^{2}$ .

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

Step 6.6

Multiply the new quotient term by the divisor.

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

Step 6.7

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{4} + 0 + 0$

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

Step 6.8

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

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

Step 6.9

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

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

$0 x$

$32$

Step 6.10

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

$\frac{x^{2}}{2}$

$0 x$

$\frac{1}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

$0 x$

$32$

Step 6.11

Multiply the new quotient term by the divisor.

$\frac{x^{2}}{2}$

$0 x$

$\frac{1}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

$0 x$

$32$

$x^{2}$

$0$

Step 6.12

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

$\frac{x^{2}}{2}$

$0 x$

$\frac{1}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

$0 x$

$32$

$x^{2}$

$0$

Step 6.13

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

$\frac{x^{2}}{2}$

$0 x$

$\frac{1}{2}$

$2 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$32$

$x^{4}$

$0$

$x^{2}$

$0 x$

$32$

$x^{2}$

$0$

$32$

Step 6.14

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

$- \frac{x^{2}}{2} + \frac{1}{2} + \frac{32}{2 x^{2}}$

Step 6.15

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

$y = - \frac{x^{2}}{2} + \frac{1}{2}$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x^{2}}{2} + \frac{1}{2}$

Step 8