Algebra Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

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

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

Step 1.2

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

$2 x = 2 \cdot 1 \cdot x$

Step 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

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

Step 2.3

Simplify.

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Step 2.3.1

One to any power is one.

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

Step 2.3.2

Multiply $x$ by $1$ .

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

Step 3

Cancel the common factor of ${(1 - x)}^{2}$ and $1 - x$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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