Pre-Algebra Examples

$\frac{(x^{n} - 3) (2 x^{n} + 5)}{x^{n - 1}}$

Step 1

Expand $(x^{n} - 3) (2 x^{n} + 5)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\frac{x^{n} (2 x^{n} + 5) - 3 (2 x^{n} + 5)}{x^{n - 1}}$

Step 1.2

Apply the distributive property.

$\frac{x^{n} (2 x^{n}) + x^{n} \cdot 5 - 3 (2 x^{n} + 5)}{x^{n - 1}}$

Step 1.3

Apply the distributive property.

$\frac{x^{n} (2 x^{n}) + x^{n} \cdot 5 - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{n} x^{n} + x^{n} \cdot 5 - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.2

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

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Step 2.1.2.1

Move $x^{n}$ .

$\frac{2 (x^{n} x^{n}) + x^{n} \cdot 5 - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.2.2

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

$\frac{2 x^{n + n} + x^{n} \cdot 5 - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.2.3

Add $n$ and $n$ .

$\frac{2 x^{2 n} + x^{n} \cdot 5 - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.3

Move $5$ to the left of $x^{n}$ .

$\frac{2 x^{2 n} + 5 \cdot x^{n} - 3 (2 x^{n}) - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.4

Multiply $2$ by $- 3$ .

$\frac{2 x^{2 n} + 5 x^{n} - 6 x^{n} - 3 \cdot 5}{x^{n - 1}}$

Step 2.1.5

Multiply $- 3$ by $5$ .

$\frac{2 x^{2 n} + 5 x^{n} - 6 x^{n} - 15}{x^{n - 1}}$

Step 2.2

Subtract $6 x^{n}$ from $5 x^{n}$ .

$\frac{2 x^{2 n} - x^{n} - 15}{x^{n - 1}}$

Step 3

Split the fraction $\frac{2 x^{2 n} - x^{n} - 15}{x^{n - 1}}$ into two fractions.

$\frac{2 x^{2 n} - x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 4

Split the fraction $\frac{2 x^{2 n} - x^{n}}{x^{n - 1}}$ into two fractions.

$\frac{2 x^{2 n}}{x^{n - 1}} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 5

Cancel the common factor of $x^{2 n}$ and $x^{n - 1}$ .

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Step 5.1

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

$\frac{x^{n - 1} (2 x^{2 n - (n - 1)})}{x^{n - 1}} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

$\frac{x^{n - 1} (2 x^{2 n - (n - 1)})}{x^{n - 1} \cdot 1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 5.2.2

Cancel the common factor.

$\frac{x^{n - 1} (2 x^{2 n - (n - 1)})}{x^{n - 1} \cdot 1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 5.2.3

Rewrite the expression.

$\frac{2 x^{2 n - (n - 1)}}{1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 5.2.4

Divide $2 x^{2 n - (n - 1)}$ by $1$ .

$2 x^{2 n - (n - 1)} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$2 x^{2 n - n - - 1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 6.2

Multiply $- 1$ by $- 1$ .

$2 x^{2 n - n + 1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 7

Subtract $n$ from $2 n$ .

$2 x^{n + 1} + \frac{- x^{n}}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 8

Cancel the common factor of $x^{n}$ and $x^{n - 1}$ .

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Step 8.1

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

$2 x^{n + 1} + \frac{x^{n - 1} (- x^{n - (n - 1)})}{x^{n - 1}} + \frac{- 15}{x^{n - 1}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Multiply by $1$ .

$2 x^{n + 1} + \frac{x^{n - 1} (- x^{n - (n - 1)})}{x^{n - 1} \cdot 1} + \frac{- 15}{x^{n - 1}}$

Step 8.2.2

Cancel the common factor.

$2 x^{n + 1} + \frac{x^{n - 1} (- x^{n - (n - 1)})}{x^{n - 1} \cdot 1} + \frac{- 15}{x^{n - 1}}$

Step 8.2.3

Rewrite the expression.

$2 x^{n + 1} + \frac{- x^{n - (n - 1)}}{1} + \frac{- 15}{x^{n - 1}}$

Step 8.2.4

Divide $- x^{n - (n - 1)}$ by $1$ .

$2 x^{n + 1} - x^{n - (n - 1)} + \frac{- 15}{x^{n - 1}}$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$2 x^{n + 1} - x^{n - n - - 1} + \frac{- 15}{x^{n - 1}}$

Step 9.2

Multiply $- 1$ by $- 1$ .

$2 x^{n + 1} - x^{n - n + 1} + \frac{- 15}{x^{n - 1}}$

Step 10

Subtract $n$ from $n$ .

$2 x^{n + 1} - x^{0 + 1} + \frac{- 15}{x^{n - 1}}$

Step 11

Add $0$ and $1$ .

$2 x^{n + 1} - x^{1} + \frac{- 15}{x^{n - 1}}$

Step 12

Simplify.

$2 x^{n + 1} - x + \frac{- 15}{x^{n - 1}}$

Step 13

Move the negative in front of the fraction.

$2 x^{n + 1} - x - \frac{15}{x^{n - 1}}$