Algebra Examples

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

Step 1

Find the common denominator.

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Step 1.1

Write $1$ as a fraction with denominator $1$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Write $x$ as a fraction with denominator $1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $x$ by $1$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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Step 3.4.1

Move $x$ .

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

Step 3.4.2

Multiply $x$ by $x$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $1 - x + x - x^{2} + x^{2}$ .

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Step 4.1.1

Add $- x$ and $x$ .

$\frac{1 + 0 - x^{2} + x^{2}}{- x + 1}$

Step 4.1.2

Add $1$ and $0$ .

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

Step 4.1.3

Add $- x^{2}$ and $x^{2}$ .

$\frac{1 + 0}{- x + 1}$

Step 4.1.4

Add $1$ and $0$ .

$\frac{1}{- x + 1}$

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Rewrite negatives.

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Step 4.5.1

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

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

Step 4.5.2

Move the negative in front of the fraction.

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