Algebra Examples

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

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{2}{x}$ by $\frac{(x - 1) (x^{2} - x)}{(x - 1) (x^{2} - x)}$ .

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

Step 1.2

Multiply $\frac{2}{x}$ by $\frac{(x - 1) (x^{2} - x)}{(x - 1) (x^{2} - x)}$ .

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

Step 1.3

Multiply $\frac{3}{x - 1}$ by $\frac{x (x^{2} - x)}{x (x^{2} - x)}$ .

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

Step 1.4

Multiply $\frac{3}{x - 1}$ by $\frac{x (x^{2} - x)}{x (x^{2} - x)}$ .

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

Step 1.5

Multiply $\frac{4}{x^{2} - x}$ by $\frac{x (x - 1)}{x (x - 1)}$ .

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

Step 1.6

Multiply $\frac{4}{x^{2} - x}$ by $\frac{x (x - 1)}{x (x - 1)}$ .

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

Step 1.7

Reorder the factors of $x ((x - 1) (x^{2} - x))$ .

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

Step 1.8

Reorder the factors of $(x - 1) (x (x^{2} - x))$ .

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

Step 1.9

Reorder the factors of $(x^{2} - x) (x (x - 1))$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify each term.

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Step 2.2.1

Expand $(x - 1) (x^{2} - x)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

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Step 2.2.2.1.1.1.1

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

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

Step 2.2.2.1.1.1.2

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

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

Step 2.2.2.1.1.2

Add $1$ and $2$ .

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

Step 2.2.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.2.1.3

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

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Step 2.2.2.1.3.1

Move $x$ .

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

Step 2.2.2.1.3.2

Multiply $x$ by $x$ .

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

Step 2.2.2.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 2.2.2.1.5

Multiply $- 1 (- x)$ .

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Step 2.2.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.1.5.2

Multiply $x$ by $1$ .

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

Step 2.2.2.2

Subtract $x^{2}$ from $- x^{2}$ .

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

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Multiply $- 2$ by $2$ .

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

Step 2.2.5

Apply the distributive property.

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

Step 2.2.6

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

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Step 2.2.6.1

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

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Step 2.2.6.1.1

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

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

Step 2.2.6.1.2

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

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

Step 2.2.6.2

Add $1$ and $2$ .

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

Step 2.2.7

Rewrite using the commutative property of multiplication.

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

Step 2.2.8

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

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Step 2.2.8.1

Move $x$ .

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

Step 2.2.8.2

Multiply $x$ by $x$ .

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

Step 2.2.9

Apply the distributive property.

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

Step 2.2.10

Multiply $- 1$ by $3$ .

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

Step 2.2.11

Apply the distributive property.

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

Step 2.2.12

Multiply $x$ by $x$ .

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

Step 2.2.13

Move $- 1$ to the left of $x$ .

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

Step 2.2.14

Rewrite $- 1 x$ as $- x$ .

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

Step 2.2.15

Apply the distributive property.

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

Step 2.2.16

Multiply $- 1$ by $- 4$ .

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

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Add $2 x^{3}$ and $3 x^{3}$ .

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

Step 2.3.2

Subtract $3 x^{2}$ from $- 4 x^{2}$ .

$\frac{5 x^{3} - 7 x^{2} + 2 x - 4 x^{2} + 4 x}{x (x - 1) (x^{2} - x)}$

Step 2.3.3

Subtract $4 x^{2}$ from $- 7 x^{2}$ .

$\frac{5 x^{3} - 11 x^{2} + 2 x + 4 x}{x (x - 1) (x^{2} - x)}$

Step 2.3.4

Add $2 x$ and $4 x$ .

$\frac{5 x^{3} - 11 x^{2} + 6 x}{x (x - 1) (x^{2} - x)}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $x$ out of $5 x^{3} - 11 x^{2} + 6 x$ .

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Step 3.1.1

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

$\frac{x (5 x^{2}) - 11 x^{2} + 6 x}{x (x - 1) (x^{2} - x)}$

Step 3.1.2

Factor $x$ out of $- 11 x^{2}$ .

$\frac{x (5 x^{2}) + x (- 11 x) + 6 x}{x (x - 1) (x^{2} - x)}$

Step 3.1.3

Factor $x$ out of $6 x$ .

$\frac{x (5 x^{2}) + x (- 11 x) + x \cdot 6}{x (x - 1) (x^{2} - x)}$

Step 3.1.4

Factor $x$ out of $x (5 x^{2}) + x (- 11 x)$ .

$\frac{x (5 x^{2} - 11 x) + x \cdot 6}{x (x - 1) (x^{2} - x)}$

Step 3.1.5

Factor $x$ out of $x (5 x^{2} - 11 x) + x \cdot 6$ .

$\frac{x (5 x^{2} - 11 x + 6)}{x (x - 1) (x^{2} - x)}$

Step 3.2

Factor by grouping.

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Step 3.2.1

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 $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = - 11$ .

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Step 3.2.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{x (5 x^{2} - 11 (x) + 6)}{x (x - 1) (x^{2} - x)}$

Step 3.2.1.2

Rewrite $- 11$ as $- 5$ plus $- 6$

$\frac{x (5 x^{2} + (- 5 - 6) x + 6)}{x (x - 1) (x^{2} - x)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{x (5 x^{2} - 5 x - 6 x + 6)}{x (x - 1) (x^{2} - x)}$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$\frac{x ((5 x^{2} - 5 x) - 6 x + 6)}{x (x - 1) (x^{2} - x)}$

Step 3.2.2.2

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

$\frac{x (5 x (x - 1) - 6 (x - 1))}{x (x - 1) (x^{2} - x)}$

Step 3.2.3

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

$\frac{x ((x - 1) (5 x - 6))}{x (x - 1) (x^{2} - x)}$

$\frac{x (x - 1) (5 x - 6)}{x (x - 1) (x^{2} - x)}$

Step 4

Simplify the denominator.

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Step 4.1

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

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Step 4.1.1

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

$\frac{x (x - 1) (5 x - 6)}{x (x - 1) (x \cdot x - x)}$

Step 4.1.2

Factor $x$ out of $- x$ .

$\frac{x (x - 1) (5 x - 6)}{x (x - 1) (x \cdot x + x \cdot - 1)}$

Step 4.1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\frac{x (x - 1) (5 x - 6)}{x (x - 1) (x (x - 1))}$

$\frac{x (x - 1) (5 x - 6)}{x (x - 1) x (x - 1)}$

Step 4.2

Combine exponents.

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Step 4.2.1

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

$\frac{x (x - 1) (5 x - 6)}{x^{1} x (x - 1) (x - 1)}$

Step 4.2.2

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

$\frac{x (x - 1) (5 x - 6)}{x^{1} x^{1} (x - 1) (x - 1)}$

Step 4.2.3

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

$\frac{x (x - 1) (5 x - 6)}{x^{1 + 1} (x - 1) (x - 1)}$

Step 4.2.4

Add $1$ and $1$ .

$\frac{x (x - 1) (5 x - 6)}{x^{2} (x - 1) (x - 1)}$

Step 4.2.5

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

$\frac{x (x - 1) (5 x - 6)}{x^{2} ({(x - 1)}^{1} (x - 1))}$

Step 4.2.6

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

$\frac{x (x - 1) (5 x - 6)}{x^{2} ({(x - 1)}^{1} {(x - 1)}^{1})}$

Step 4.2.7

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

$\frac{x (x - 1) (5 x - 6)}{x^{2} {(x - 1)}^{1 + 1}}$

Step 4.2.8

Add $1$ and $1$ .

$\frac{x (x - 1) (5 x - 6)}{x^{2} {(x - 1)}^{2}}$

Step 5

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

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Step 5.1

Factor $x$ out of $x (x - 1) (5 x - 6)$ .

$\frac{x ((x - 1) (5 x - 6))}{x^{2} {(x - 1)}^{2}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\frac{x ((x - 1) (5 x - 6))}{x (x {(x - 1)}^{2})}$

Step 5.2.2

Cancel the common factor.

$\frac{x ((x - 1) (5 x - 6))}{x (x {(x - 1)}^{2})}$

Step 5.2.3

Rewrite the expression.

$\frac{(x - 1) (5 x - 6)}{x {(x - 1)}^{2}}$

Step 6

Cancel the common factors.

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Step 6.1

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

$\frac{(x - 1) (5 x - 6)}{(x - 1) (x (x - 1))}$

Step 6.2

Cancel the common factor.

$\frac{(x - 1) (5 x - 6)}{(x - 1) (x (x - 1))}$

Step 6.3

Rewrite the expression.

$\frac{5 x - 6}{x (x - 1)}$