Algebra Examples

$\cot^{3} (x) + \cot^{2} (x) + \cot (x) + 1$

Step 1

Regroup terms.

$\cot^{3} (x) + 1 + \cot^{2} (x) + \cot (x)$

Step 2

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

$\cot^{3} (x) + 1^{3} + \cot^{2} (x) + \cot (x)$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $1$ .

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

Step 4.2

One to any power is one.

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

Step 5

Factor $\cot (x)$ out of $\cot^{2} (x) + \cot (x)$ .

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Step 5.1

Factor $\cot (x)$ out of $\cot^{2} (x)$ .

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

Step 5.2

Multiply by $1$ .

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

Step 5.3

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

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

Step 6

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

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Step 6.1

Factor $\cot (x) + 1$ out of $\cot (x) (\cot (x) + 1)$ .

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

Step 6.2

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

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

Step 7

Add $- \cot (x)$ and $\cot (x)$ .

$(\cot (x) + 1) (\cot^{2} (x) + 0 + 1)$

Step 8

Add $\cot^{2} (x)$ and $0$ .

$(\cot (x) + 1) (\cot^{2} (x) + 1)$