Pre-Algebra Examples

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

Step 1

To write $\frac{x}{x^{2} - 1}$ as a fraction with a common denominator, multiply by $\frac{x - x^{2}}{x - x^{2}}$ .

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

Step 2

To write $\frac{5 x}{x - x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 1}{x^{2} - 1}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $\frac{5 x}{x - x^{2}}$ by $\frac{x^{2} - 1}{x^{2} - 1}$ .

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Rewrite $\frac{x (x - x^{2}) + 5 x (x^{2} - 1)}{(x - x^{2}) (x^{2} - 1)}$ in a factored form.

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Step 5.1

Factor $x$ out of $x (x - x^{2}) + 5 x (x^{2} - 1)$ .

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Step 5.1.1

Factor $x$ out of $5 x (x^{2} - 1)$ .

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

Step 5.1.2

Factor $x$ out of $x (x - x^{2}) + x (5 (x^{2} - 1))$ .

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

Step 5.2

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 5.3

Factor by grouping.

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Step 5.3.1

Reorder terms.

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

Step 5.3.2

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 $a \cdot c = 4 \cdot - 5 = - 20$ and whose sum is $b = 1$ .

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Step 5.3.2.1

Multiply by $1$ .

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

Step 5.3.2.2

Rewrite $1$ as $- 4$ plus $5$

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

Step 5.3.2.3

Apply the distributive property.

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

Step 5.3.3

Factor out the greatest common factor from each group.

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Step 5.3.3.1

Group the first two terms and the last two terms.

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

Step 5.3.3.2

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

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

Step 5.3.4

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

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

Step 5.4

Factor.

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Step 5.4.1

Replace all occurrences of $u$ with $x$ .

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

Step 5.4.2

Remove unnecessary parentheses.

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

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

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

Step 5.5.5

Multiply $x$ by $1 - x$ .

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

Step 5.6

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

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

Step 5.7

Factor.

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Step 5.7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 5.7.2

Remove unnecessary parentheses.

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

Step 5.8

Combine exponents.

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Step 5.8.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.8.2

Factor $- 1$ out of $- x$ .

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

Step 5.8.3

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

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

Step 5.8.4

Remove parentheses.

$\frac{x (x - 1) (4 x + 5)}{x \cdot - 1 (- 1 + x) (x + 1) (x - 1)}$

Step 5.8.5

Reorder terms.

$\frac{x (x - 1) (4 x + 5)}{x \cdot - 1 (x - 1) (x + 1) (x - 1)}$

Step 5.8.6

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

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

Step 5.8.7

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

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

Step 5.8.8

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

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

Step 5.8.9

Add $1$ and $1$ .

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

Step 5.9

Reduce the expression $\frac{x (x - 1) (4 x + 5)}{x \cdot - 1 {(x - 1)}^{2} (x + 1)}$ by cancelling the common factors.

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Step 5.9.1

Cancel the common factor.

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

Step 5.9.2

Rewrite the expression.

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

Step 6

Reduce the expression $\frac{(x - 1) (4 x + 5)}{- 1 {(x - 1)}^{2} (x + 1)}$ by cancelling the common factors.

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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