Pre-Algebra Examples

$p x^{7} - x^{5} - 5 x + 2$

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

Add $x^{5}$ to both sides of the equation.

$y x^{7} - 5 x + 2 = x^{5}$

Step 1.1.2

Add $5 x$ to both sides of the equation.

$y x^{7} + 2 = x^{5} + 5 x$

Step 1.1.3

Subtract $2$ from both sides of the equation.

$y x^{7} = x^{5} + 5 x - 2$

Step 1.2

Divide each term in $y x^{7} = x^{5} + 5 x - 2$ by $x^{7}$ and simplify.

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Step 1.2.1

Divide each term in $y x^{7} = x^{5} + 5 x - 2$ by $x^{7}$ .

$\frac{y x^{7}}{x^{7}} = \frac{x^{5}}{x^{7}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $x^{7}$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{y x^{7}}{x^{7}} = \frac{x^{5}}{x^{7}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{5}}{x^{7}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

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

Cancel the common factor of $x^{5}$ and $x^{7}$ .

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Step 1.2.3.1.1.1

Multiply by $1$ .

$y = \frac{x^{5} \cdot 1}{x^{7}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $x^{5}$ out of $x^{7}$ .

$y = \frac{x^{5} \cdot 1}{x^{5} x^{2}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{5} \cdot 1}{x^{5} x^{2}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{x^{2}} + \frac{5 x}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.2

Cancel the common factor of $x$ and $x^{7}$ .

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Step 1.2.3.1.2.1

Factor $x$ out of $5 x$ .

$y = \frac{1}{x^{2}} + \frac{x \cdot 5}{x^{7}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $x$ out of $x^{7}$ .

$y = \frac{1}{x^{2}} + \frac{x \cdot 5}{x \cdot x^{6}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{1}{x^{2}} + \frac{x \cdot 5}{x \cdot x^{6}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{1}{x^{2}} + \frac{5}{x^{6}} + \frac{- 2}{x^{7}}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{1}{x^{2}} + \frac{5}{x^{6}} - \frac{2}{x^{7}}$

Step 2

Find where the expression $\frac{1}{x^{2}} + \frac{5}{x^{6}} - \frac{2}{x^{7}}$ 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 $m$ .

$n = 5$

$m = 7$

Step 5

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

$y = 0$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

No Oblique Asymptotes

Step 8