Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $- 3$ out of $- 3 x - \frac{3}{x}$ .

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Step 2.1

Factor $- 3$ out of $- 3 x$ .

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

Step 2.2

Factor $- 3$ out of $- \frac{3}{x}$ .

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

Step 2.3

Factor $- 3$ out of $- 3 (x) - 3 (- \frac{- 1}{x})$ .

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

Step 3

Move the negative in front of the fraction.

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

Step 4

Multiply $- - \frac{1}{x}$ .

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\frac{1}{x}$ by $1$ .

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

Step 5

Factor using the perfect square rule.

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Step 5.1

Rearrange terms.

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

Step 5.2

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

$2 = 2 \cdot x \cdot \frac{1}{x}$

Step 5.3

Rewrite the polynomial.

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

Step 5.4

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

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

Step 6

Factor $x + \frac{1}{x}$ out of $- 3 (x + \frac{1}{x}) + {(x + \frac{1}{x})}^{2}$ .

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Step 6.1

Factor $x + \frac{1}{x}$ out of $- 3 (x + \frac{1}{x})$ .

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

Step 6.2

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

$(x + \frac{1}{x}) \cdot - 3 + (x + \frac{1}{x}) (x + \frac{1}{x})$

Step 6.3

Factor $x + \frac{1}{x}$ out of $(x + \frac{1}{x}) \cdot - 3 + (x + \frac{1}{x}) (x + \frac{1}{x})$ .

$(x + \frac{1}{x}) (- 3 + x + \frac{1}{x})$

Step 7

Reorder terms.

$(x + \frac{1}{x}) (x + \frac{1}{x} - 3)$