Álgebra Ejemplos

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

Paso 1

Reagrupa los términos.

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

Paso 2

Reescribe $1$ como $1^{3}$ .

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

Paso 3

Dado que ambos términos son cubos perfectos, factoriza con la fórmula de la suma de cubos, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ , donde $a = \cot (x)$ y $b = 1$ .

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

Paso 4

Simplifica.

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Paso 4.1

Multiplica $- 1$ por $1$ .

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

Paso 4.2

Uno elevado a cualquier potencia es uno.

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

Paso 5

Factoriza $\cot (x)$ de $\cot^{2} (x) + \cot (x)$ .

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Paso 5.1

Factoriza $\cot (x)$ de $\cot^{2} (x)$ .

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

Paso 5.2

Multiplica por $1$ .

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

Paso 5.3

Factoriza $\cot (x)$ de $\cot (x) \cot (x) + \cot (x) \cdot 1$ .

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

Paso 6

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

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Paso 6.1

Factoriza $\cot (x) + 1$ de $\cot (x) (\cot (x) + 1)$ .

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

Paso 6.2

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

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

Paso 7

Suma $- \cot (x)$ y $\cot (x)$ .

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

Paso 8

Suma $\cot^{2} (x)$ y $0$ .

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