Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the numerator.

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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.

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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

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 $b = 1$ .

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

Step 2.5

Combine exponents.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Add $1$ and $1$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Divide $x - 1$ by $1$ .

$x - 1$