Finite Math Examples

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} = 2$

Step 1

Multiply both sides by $x^{2} + x - 2$ .

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} (x^{2} + x - 2) = 2 (x^{2} + x - 2)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} (x^{2} + x - 2)$ .

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Step 2.1.1.1

Cancel the common factor of $x^{2} + x - 2$ .

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Step 2.1.1.1.1

Cancel the common factor.

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} (x^{2} + x - 2) = 2 (x^{2} + x - 2)$

Step 2.1.1.1.2

Rewrite the expression.

$x^{5} - 3 x^{3} + p x^{3} + q = 2 (x^{2} + x - 2)$

Step 2.1.1.2

Move $- 3 x^{3}$ .

$x^{5} + p x^{3} - 3 x^{3} + q = 2 (x^{2} + x - 2)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (x^{2} + x - 2)$ .

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Step 2.2.1.1

Apply the distributive property.

$x^{5} + p x^{3} - 3 x^{3} + q = 2 x^{2} + 2 x + 2 \cdot - 2$

Step 2.2.1.2

Multiply $2$ by $- 2$ .

$x^{5} + p x^{3} - 3 x^{3} + q = 2 x^{2} + 2 x - 4$

Step 3

Solve for $p$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x^{5}$ from both sides of the equation.

$p x^{3} - 3 x^{3} + q = 2 x^{2} + 2 x - 4 - x^{5}$

Step 3.1.2

Add $3 x^{3}$ to both sides of the equation.

$p x^{3} + q = 2 x^{2} + 2 x - 4 - x^{5} + 3 x^{3}$

Step 3.1.3

Subtract $q$ from both sides of the equation.

$p x^{3} = 2 x^{2} + 2 x - 4 - x^{5} + 3 x^{3} - q$

Step 3.2

Divide each term in $p x^{3} = 2 x^{2} + 2 x - 4 - x^{5} + 3 x^{3} - q$ by $x^{3}$ and simplify.

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Step 3.2.1

Divide each term in $p x^{3} = 2 x^{2} + 2 x - 4 - x^{5} + 3 x^{3} - q$ by $x^{3}$ .

$\frac{p x^{3}}{x^{3}} = \frac{2 x^{2}}{x^{3}} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x^{3}$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{p x^{3}}{x^{3}} = \frac{2 x^{2}}{x^{3}} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.2.1.2

Divide $p$ by $1$ .

$p = \frac{2 x^{2}}{x^{3}} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.2.3.1.1.1

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

$p = \frac{x^{2} \cdot 2}{x^{3}} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$p = \frac{x^{2} \cdot 2}{x^{2} x} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$p = \frac{x^{2} \cdot 2}{x^{2} x} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$p = \frac{2}{x} + \frac{2 x}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.2.3.1.2.1

Factor $x$ out of $2 x$ .

$p = \frac{2}{x} + \frac{x \cdot 2}{x^{3}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $x$ out of $x^{3}$ .

$p = \frac{2}{x} + \frac{x \cdot 2}{x \cdot x^{2}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$p = \frac{2}{x} + \frac{x \cdot 2}{x \cdot x^{2}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$p = \frac{2}{x} + \frac{2}{x^{2}} + \frac{- 4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.3

Move the negative in front of the fraction.

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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Step 3.2.3.1.4.1

Factor $x^{3}$ out of $- x^{5}$ .

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} + \frac{x^{3} (- x^{2})}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4.2

Cancel the common factors.

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Step 3.2.3.1.4.2.1

Multiply by $1$ .

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} + \frac{x^{3} (- x^{2})}{x^{3} \cdot 1} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4.2.2

Cancel the common factor.

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} + \frac{x^{3} (- x^{2})}{x^{3} \cdot 1} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4.2.3

Rewrite the expression.

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} + \frac{- x^{2}}{1} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4.2.4

Divide $- x^{2}$ by $1$ .

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} - x^{2} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.5

Cancel the common factor of $x^{3}$ .

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Step 3.2.3.1.5.1

Cancel the common factor.

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} - x^{2} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.5.2

Divide $3$ by $1$ .

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} - x^{2} + 3 + \frac{- q}{x^{3}}$

Step 3.2.3.1.6

Move the negative in front of the fraction.

$p = \frac{2}{x} + \frac{2}{x^{2}} - \frac{4}{x^{3}} - x^{2} + 3 - \frac{q}{x^{3}}$