Finite Math Examples

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} = 1$

Step 1

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

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} (x^{2} + x - 2) = 1 (x^{2} + x - 2)$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

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

Tap for more steps...

Step 2.1.1.1.1

Cancel the common factor.

$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2} (x^{2} + x - 2) = 1 (x^{2} + x - 2)$

Step 2.1.1.1.2

Rewrite the expression.

$x^{5} - 3 x^{3} + p x^{3} + q = 1 (x^{2} + x - 2)$

Step 2.1.1.2

Move $- 3 x^{3}$ .

$x^{5} + p x^{3} - 3 x^{3} + q = 1 (x^{2} + x - 2)$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

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

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

Step 3

Solve for $p$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Subtract $q$ from both sides of the equation.

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

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

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $p$ by $1$ .

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

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

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

Tap for more steps...

Step 3.2.3.1.1.1

Multiply by $1$ .

$p = \frac{x^{2} \cdot 1}{x^{3}} + \frac{x}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.1.2.1

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

$p = \frac{x^{2} \cdot 1}{x^{2} x} + \frac{x}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$p = \frac{x^{2} \cdot 1}{x^{2} x} + \frac{x}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$p = \frac{1}{x} + \frac{x}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2

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

Tap for more steps...

Step 3.2.3.1.2.1

Raise $x$ to the power of $1$ .

$p = \frac{1}{x} + \frac{x^{1}}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.2

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

$p = \frac{1}{x} + \frac{x \cdot 1}{x^{3}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.3

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.2.3.1

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

$p = \frac{1}{x} + \frac{x \cdot 1}{x \cdot x^{2}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.3.2

Cancel the common factor.

$p = \frac{1}{x} + \frac{x \cdot 1}{x \cdot x^{2}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.2.3.3

Rewrite the expression.

$p = \frac{1}{x} + \frac{1}{x^{2}} + \frac{- 2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.3

Move the negative in front of the fraction.

$p = \frac{1}{x} + \frac{1}{x^{2}} - \frac{2}{x^{3}} + \frac{- x^{5}}{x^{3}} + \frac{3 x^{3}}{x^{3}} + \frac{- q}{x^{3}}$

Step 3.2.3.1.4

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

Tap for more steps...

Step 3.2.3.1.4.1

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

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

Step 3.2.3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.4.2.1

Multiply by $1$ .

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

Step 3.2.3.1.4.2.2

Cancel the common factor.

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

Step 3.2.3.1.4.2.3

Rewrite the expression.

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

Step 3.2.3.1.4.2.4

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

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

Step 3.2.3.1.5

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

Tap for more steps...

Step 3.2.3.1.5.1

Cancel the common factor.

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

Step 3.2.3.1.5.2

Divide $3$ by $1$ .

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

Step 3.2.3.1.6

Move the negative in front of the fraction.

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