Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $4$ out of $4 x^{2} + 4$ .

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Step 1.1.1

Factor $4$ out of $4 x^{2}$ .

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

Step 1.1.2

Factor $4$ out of $4$ .

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

Step 1.1.3

Factor $4$ out of $4 (x^{2}) + 4 (1)$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

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

Step 1.2.4.2

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 $b = 1$ .

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

Step 2

To write $\frac{3 x}{4 (x^{2} + 1)}$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

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

Step 3

To write $- \frac{2 x^{2}}{(x^{2} + 1) (x + 1) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 4

Write each expression with a common denominator of $4 (x^{2} + 1) (x + 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $4 (x^{2} + 1) ((x + 1) (x - 1))$ .

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

Step 4.4

Reorder the factors of $(x^{2} + 1) (x + 1) (x - 1) \cdot 4$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $3 x ((x + 1) (x - 1)) - 2 x^{2} \cdot 4$ .

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Step 6.1.1

Factor $x$ out of $3 x ((x + 1) (x - 1))$ .

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

Step 6.1.2

Factor $x$ out of $- 2 x^{2} \cdot 4$ .

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

Step 6.1.3

Factor $x$ out of $x (3 ((x + 1) (x - 1))) + x (- 2 x \cdot 4)$ .

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

Step 6.2

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $x$ by $x$ .

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

Step 6.3.1.2

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

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

Step 6.3.1.3

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

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

Step 6.3.1.4

Multiply $x$ by $1$ .

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

Step 6.3.1.5

Multiply $- 1$ by $1$ .

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

Step 6.3.2

Add $- x$ and $x$ .

$\frac{x (3 (x^{2} + 0 - 1) - 2 x \cdot 4)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.3.3

Add $x^{2}$ and $0$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $3$ by $- 1$ .

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

Step 6.6

Multiply $4$ by $- 2$ .

$\frac{x (3 x^{2} - 3 - 8 x)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.7

Reorder terms.

$\frac{x (3 x^{2} - 8 x - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8

Factor by grouping.

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Step 6.8.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

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Step 6.8.1.1

Factor $- 8$ out of $- 8 x$ .

$\frac{x (3 x^{2} - 8 (x) - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8.1.2

Rewrite $- 8$ as $1$ plus $- 9$

$\frac{x (3 x^{2} + (1 - 9) x - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8.1.3

Apply the distributive property.

$\frac{x (3 x^{2} + 1 x - 9 x - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8.1.4

Multiply $x$ by $1$ .

$\frac{x (3 x^{2} + x - 9 x - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8.2

Factor out the greatest common factor from each group.

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Step 6.8.2.1

Group the first two terms and the last two terms.

$\frac{x ((3 x^{2} + x) - 9 x - 3)}{4 (x + 1) (x - 1) (x^{2} + 1)}$

Step 6.8.2.2

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

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

Step 6.8.3

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

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

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