Precalculus Examples

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

Step 1

Regroup terms.

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

Step 2

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

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

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 $a = \csc (x)$ and $b = 1$ .

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $1$ .

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

Step 4.2

One to any power is one.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Factor $\csc (x)$ out of $- \csc (x)$ .

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

Step 5.3

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

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

Step 6

Factor.

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Step 6.1

Factor $- 1$ out of $- \csc (x) - 1$ .

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Step 6.1.1

Factor $- 1$ out of $- \csc (x)$ .

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

Step 6.1.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 6.1.3

Factor $- 1$ out of $- (\csc (x)) - 1 (1)$ .

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

Step 6.2

Remove unnecessary parentheses.

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

Step 7

Factor out negative.

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

Step 8

Remove parentheses.

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

Step 9

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

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Step 9.1

Factor $\csc (x) + 1$ out of $- \csc (x) (\csc (x) + 1)$ .

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

Step 9.2

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

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

Step 10

Subtract $\csc (x)$ from $- \csc (x)$ .

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

Step 11

Factor using the perfect square rule.

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Step 11.1

Rearrange terms.

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

Step 11.2

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

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

Step 11.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \csc (x) = 2 \cdot \csc (x) \cdot 1$

Step 11.4

Rewrite the polynomial.

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

Step 11.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = \csc (x)$ and $b = 1$ .

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