Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

$\frac{a \cdot 1 - a^{4}}{a^{2} + a + 1}$

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.2

Rewrite $1$ as $1^{3}$ .

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

Step 1.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = a$ .

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

Step 1.4

Simplify.

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Step 1.4.1

One to any power is one.

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

Step 1.4.2

Multiply $a$ by $1$ .

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

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

Step 2

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

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Step 2.1

Reorder terms.

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

Step 2.2

Cancel the common factor.

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

Step 2.3

Divide $a (1 - a)$ by $1$ .

$a (1 - a)$

Step 3

Apply the distributive property.

$a \cdot 1 + a (- a)$

Step 4

Multiply $a$ by $1$ .

$a + a (- a)$

Step 5

Rewrite using the commutative property of multiplication.

$a - a \cdot a$

Step 6

Multiply $a$ by $a$ by adding the exponents.

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Step 6.1

Move $a$ .

$a - (a \cdot a)$

Step 6.2

Multiply $a$ by $a$ .

$a - a^{2}$