Pre-Algebra Examples

$\frac{5}{x} - \frac{10}{x^{2}} + \frac{1}{x} - \frac{4}{x^{3}}$

Step 1

Combine fractions.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{5 + 1}{x} + \frac{- 10}{x^{2}} + \frac{- 4}{x^{3}}$

Step 1.2

Add $5$ and $1$ .

$\frac{6}{x} + \frac{- 10}{x^{2}} + \frac{- 4}{x^{3}}$

Step 2

Simplify each term.

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Step 2.1

Move the negative in front of the fraction.

$\frac{6}{x} - \frac{10}{x^{2}} + \frac{- 4}{x^{3}}$

Step 2.2

Move the negative in front of the fraction.

$\frac{6}{x} - \frac{10}{x^{2}} - \frac{4}{x^{3}}$

Step 3

To write $\frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{6}{x} \cdot \frac{x}{x} - \frac{10}{x^{2}} - \frac{4}{x^{3}}$

Step 4

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

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Step 4.1

Multiply $\frac{6}{x}$ by $\frac{x}{x}$ .

$\frac{6 x}{x \cdot x} - \frac{10}{x^{2}} - \frac{4}{x^{3}}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{6 x}{x^{2}} - \frac{10}{x^{2}} - \frac{4}{x^{3}}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 x - 10}{x^{2}} - \frac{4}{x^{3}}$

Step 6

Factor $2$ out of $6 x - 10$ .

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Step 6.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x) - 10}{x^{2}} - \frac{4}{x^{3}}$

Step 6.2

Factor $2$ out of $- 10$ .

$\frac{2 (3 x) + 2 (- 5)}{x^{2}} - \frac{4}{x^{3}}$

Step 6.3

Factor $2$ out of $2 (3 x) + 2 (- 5)$ .

$\frac{2 (3 x - 5)}{x^{2}} - \frac{4}{x^{3}}$

Step 7

To write $\frac{2 (3 x - 5)}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 (3 x - 5)}{x^{2}} \cdot \frac{x}{x} - \frac{4}{x^{3}}$

Step 8

Write each expression with a common denominator of $x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{2 (3 x - 5)}{x^{2}}$ by $\frac{x}{x}$ .

$\frac{2 (3 x - 5) x}{x^{2} x} - \frac{4}{x^{3}}$

Step 8.2

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

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Step 8.2.1

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

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Step 8.2.1.1

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

$\frac{2 (3 x - 5) x}{x^{2} x^{1}} - \frac{4}{x^{3}}$

Step 8.2.1.2

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

$\frac{2 (3 x - 5) x}{x^{2 + 1}} - \frac{4}{x^{3}}$

Step 8.2.2

Add $2$ and $1$ .

$\frac{2 (3 x - 5) x}{x^{3}} - \frac{4}{x^{3}}$

Step 9

Combine the numerators over the common denominator.

$\frac{2 (3 x - 5) x - 4}{x^{3}}$

Step 10

Simplify the numerator.

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Step 10.1

Factor $2$ out of $2 (3 x - 5) x - 4$ .

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Step 10.1.1

Factor $2$ out of $2 (3 x - 5) x$ .

$\frac{2 ((3 x - 5) x) - 4}{x^{3}}$

Step 10.1.2

Factor $2$ out of $- 4$ .

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

Step 10.1.3

Factor $2$ out of $2 ((3 x - 5) x) + 2 \cdot - 2$ .

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

Step 10.2

Apply the distributive property.

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

Step 10.3

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

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Step 10.3.1

Move $x$ .

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

Step 10.3.2

Multiply $x$ by $x$ .

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

Step 10.4

Factor by grouping.

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Step 10.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

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Step 10.4.1.1

Factor $- 5$ out of $- 5 x$ .

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

Step 10.4.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{2 (3 x^{2} + (1 - 6) x - 2)}{x^{3}}$

Step 10.4.1.3

Apply the distributive property.

$\frac{2 (3 x^{2} + 1 x - 6 x - 2)}{x^{3}}$

Step 10.4.1.4

Multiply $x$ by $1$ .

$\frac{2 (3 x^{2} + x - 6 x - 2)}{x^{3}}$

Step 10.4.2

Factor out the greatest common factor from each group.

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Step 10.4.2.1

Group the first two terms and the last two terms.

$\frac{2 ((3 x^{2} + x) - 6 x - 2)}{x^{3}}$

Step 10.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 10.4.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

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