Algebra Examples

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

Step 1

Factor $x^{2} + x - 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

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

Step 2.1.2

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

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

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine exponents.

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Step 2.5.1

Raise $x - 1$ to the power of $1$ .

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

Step 2.5.2

Raise $x - 1$ to the power of $1$ .

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

Step 2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.5.4

Add $1$ and $1$ .

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

Step 3

Cancel the common factors.

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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