Finite Math Examples

$\frac{1 - x - \frac{2}{x}}{\frac{6}{x^{2}} + \frac{1}{x} - 1}$

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

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Step 1.1

Multiply $\frac{1 - x - \frac{2}{x}}{\frac{6}{x^{2}} + \frac{1}{x} - 1}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{1 - x - \frac{2}{x}}{\frac{6}{x^{2}} + \frac{1}{x} - 1}$

Step 1.2

Combine.

$\frac{x^{2} (1 - x - \frac{2}{x})}{x^{2} (\frac{6}{x^{2}} + \frac{1}{x} - 1)}$

Step 2

Apply the distributive property.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x^{2} (- \frac{2}{x})}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Move the leading negative in $- \frac{2}{x}$ into the numerator.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x^{2} \frac{- 2}{x}}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.1.2

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot x \frac{- 2}{x}}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.1.3

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot x \frac{- 2}{x}}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.1.4

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{x^{2} \frac{6}{x^{2}} + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.2.2

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{6 + x^{2} \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{6 + x \cdot x \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.3.2

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{6 + x \cdot x \frac{1}{x} + x^{2} \cdot - 1}$

Step 3.3.3

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x^{2} (- x) + x \cdot - 2}{6 + x + x^{2} \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Factor $x$ out of $x^{2} \cdot 1$ .

$\frac{x (x \cdot 1) + x^{2} (- x) + x \cdot - 2}{6 + x + x^{2} \cdot - 1}$

Step 4.1.2

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

$\frac{x (x \cdot 1) + x (x (- x)) + x \cdot - 2}{6 + x + x^{2} \cdot - 1}$

Step 4.1.3

Factor $x$ out of $x \cdot - 2$ .

$\frac{x (x \cdot 1) + x (x (- x)) + x (- 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.1.4

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

$\frac{x (x \cdot 1 + x (- x)) + x (- 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.1.5

Factor $x$ out of $x (x \cdot 1 + x (- x)) + x (- 2)$ .

$\frac{x (x \cdot 1 + x (- x) - 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.2

Multiply $x$ by $1$ .

$\frac{x (x + x (- x) - 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.3

Rewrite using the commutative property of multiplication.

$\frac{x (x - x \cdot x - 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.4.1

Move $x$ .

$\frac{x (x - (x \cdot x) - 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.4.2

Multiply $x$ by $x$ .

$\frac{x (x - x^{2} - 2)}{6 + x + x^{2} \cdot - 1}$

Step 4.5

Reorder terms.

$\frac{x (- x^{2} + x - 2)}{6 + x + x^{2} \cdot - 1}$

Step 5

Factor by grouping.

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Step 5.1

Reorder terms.

$\frac{x (- x^{2} + x - 2)}{- x^{2} + x + 6}$

Step 5.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 1$ .

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Step 5.2.1

Multiply by $1$ .

$\frac{x (- x^{2} + x - 2)}{- x^{2} + 1 x + 6}$

Step 5.2.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{x (- x^{2} + x - 2)}{- x^{2} + (- 2 + 3) x + 6}$

Step 5.2.3

Apply the distributive property.

$\frac{x (- x^{2} + x - 2)}{- x^{2} - 2 x + 3 x + 6}$

Step 5.3

Factor out the greatest common factor from each group.

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Step 5.3.1

Group the first two terms and the last two terms.

$\frac{x (- x^{2} + x - 2)}{(- x^{2} - 2 x) + 3 x + 6}$

Step 5.3.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.4

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

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

Step 6

Factor $- 1$ out of $- x^{2}$ .

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

Step 7

Factor $- 1$ out of $x$ .

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

Step 8

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

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

Step 9

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 10

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

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

Step 11

Rewrite $- (x^{2} - x + 2)$ as $- 1 (x^{2} - x + 2)$ .

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

Step 12

Factor $- 1$ out of $- x$ .

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

Step 13

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 14

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 15

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

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

Step 16

Cancel the common factor.

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

Step 17

Rewrite the expression.

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