Algebra Examples

$a^{7} b^{2} - a b^{2}$

Step 1

Factor $a b^{2}$ out of $a^{7} b^{2} - a b^{2}$ .

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Step 1.1

Factor $a b^{2}$ out of $a^{7} b^{2}$ .

$a b^{2} (a^{6}) - a b^{2}$

Step 1.2

Factor $a b^{2}$ out of $- a b^{2}$ .

$a b^{2} (a^{6}) + a b^{2} (- 1)$

Step 1.3

Factor $a b^{2}$ out of $a b^{2} (a^{6}) + a b^{2} (- 1)$ .

$a b^{2} (a^{6} - 1)$

Step 2

Rewrite $a^{6}$ as ${(a^{2})}^{3}$ .

$a b^{2} ({(a^{2})}^{3} - 1)$

Step 3

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

$a b^{2} ({(a^{2})}^{3} - 1^{3})$

Step 4

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 $a = a^{2}$ and $b = 1$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Multiply $a^{2}$ by $1$ .

$a b^{2} ((a + 1) (a - 1) ({(a^{2})}^{2} + a^{2} + 1^{2}))$

Step 5.2

Remove unnecessary parentheses.

$a b^{2} (a + 1) (a - 1) ({(a^{2})}^{2} + a^{2} + 1^{2})$

Step 6

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.2

Multiply $2$ by $2$ .

$a b^{2} (a + 1) (a - 1) (a^{4} + a^{2} + 1^{2})$

Step 7

One to any power is one.

$a b^{2} (a + 1) (a - 1) (a^{4} + a^{2} + 1)$

Step 8

Factor.

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Step 8.1

Rewrite $a^{4} + a^{2} + 1$ in a factored form.

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Step 8.1.1

Rewrite the middle term.

$a b^{2} (a + 1) (a - 1) (a^{4} + 2 a^{2} \cdot 1 - a^{2} + 1)$

Step 8.1.2

Rearrange terms.

$a b^{2} (a + 1) (a - 1) (a^{4} + 2 a^{2} \cdot 1 + 1 - a^{2})$

Step 8.1.3

Factor first three terms by perfect square rule.

$a b^{2} (a + 1) (a - 1) ({(a^{2} + 1)}^{2} - a^{2})$

Step 8.1.4

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

$a b^{2} (a + 1) (a - 1) ((a^{2} + 1 + a) (a^{2} + 1 - a))$

Step 8.1.5

Simplify.

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Step 8.1.5.1

Reorder terms.

$a b^{2} (a + 1) (a - 1) ((a^{2} + a + 1) (a^{2} + 1 - a))$

Step 8.1.5.2

Reorder terms.

$a b^{2} (a + 1) (a - 1) ((a^{2} + a + 1) (a^{2} - a + 1))$

Step 8.2

Remove unnecessary parentheses.

$a b^{2} (a + 1) (a - 1) (a^{2} + a + 1) (a^{2} - a + 1)$