Algebra Examples

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

Step 1

Combine fractions.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Subtract $8$ from $4$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $4$ out of $- 4$ .

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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 $b = 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

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 $b = 1$ .

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

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x^{2} + 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Divide $(4 (x + 1)) (x - 1)$ by $1$ .

$(4 (x + 1)) (x - 1)$

Step 3.2

Apply the distributive property.

$(4 x + 4 \cdot 1) (x - 1)$

Step 3.3

Multiply $4$ by $1$ .

$(4 x + 4) (x - 1)$

Step 4

Expand $(4 x + 4) (x - 1)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$4 x (x - 1) + 4 (x - 1)$

Step 4.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 4 (x - 1)$

Step 4.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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Step 5.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 5.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 5.1.2

Multiply $- 1$ by $4$ .

$4 x^{2} - 4 x + 4 x + 4 \cdot - 1$

Step 5.1.3

Multiply $4$ by $- 1$ .

$4 x^{2} - 4 x + 4 x - 4$

Step 5.2

Add $- 4 x$ and $4 x$ .

$4 x^{2} + 0 - 4$

Step 5.3

Add $4 x^{2}$ and $0$ .

$4 x^{2} - 4$