Algebra Examples

$(\frac{1}{(x - 1)} - \frac{3 x}{x^{3} - 1} - \frac{3}{x^{2} + x + 1}) \div \frac{x^{2} - 5 x + 4}{x^{2} + x + 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$(\frac{1}{x - 1} - \frac{3 x}{x^{3} - 1} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{x^{2} - 5 x + 4}$

Step 2

Factor $x^{2} - 5 x + 4$ using the AC method.

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Step 2.1

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 2.2

Write the factored form using these integers.

$(\frac{1}{x - 1} - \frac{3 x}{x^{3} - 1} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 3

Simplify the denominator.

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Step 3.1

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

$(\frac{1}{x - 1} - \frac{3 x}{x^{3} - 1^{3}} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 3.2

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 $a = x$ and $b = 1$ .

$(\frac{1}{x - 1} - \frac{3 x}{(x - 1) (x^{2} + x \cdot 1 + 1^{2})} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 3.3

Simplify.

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Step 3.3.1

Multiply $x$ by $1$ .

$(\frac{1}{x - 1} - \frac{3 x}{(x - 1) (x^{2} + x + 1^{2})} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 3.3.2

One to any power is one.

$(\frac{1}{x - 1} - \frac{3 x}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 4

To write $\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + x + 1}{x^{2} + x + 1}$ .

$(\frac{1}{x - 1} \cdot \frac{x^{2} + x + 1}{x^{2} + x + 1} - \frac{3 x}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 5

Simplify terms.

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Step 5.1

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

$(\frac{x^{2} + x + 1}{(x - 1) (x^{2} + x + 1)} - \frac{3 x}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 5.2

Combine the numerators over the common denominator.

$(\frac{x^{2} + x + 1 - 3 x}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 5.3

Subtract $3 x$ from $x$ .

$(\frac{x^{2} - 2 x + 1}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6

Simplify each term.

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Step 6.1

Factor using the perfect square rule.

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Step 6.1.1

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

$(\frac{x^{2} - 2 x + 1^{2}}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6.1.2

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

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

Step 6.1.3

Rewrite the polynomial.

$(\frac{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6.1.4

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

$(\frac{{(x - 1)}^{2}}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6.2

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

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Step 6.2.1

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

$(\frac{(x - 1) (x - 1)}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Cancel the common factor.

$(\frac{(x - 1) (x - 1)}{(x - 1) (x^{2} + x + 1)} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 6.2.2.2

Rewrite the expression.

$(\frac{x - 1}{x^{2} + x + 1} - \frac{3}{x^{2} + x + 1}) \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 7

Simplify terms.

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Step 7.1

Combine the numerators over the common denominator.

$\frac{x - 1 - 3}{x^{2} + x + 1} \cdot \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 7.2

Subtract $3$ from $- 1$ .

$\frac{x - 4}{x^{2} + x + 1} \cdot \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 7.3

Cancel the common factor of $x - 4$ .

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Step 7.3.1

Cancel the common factor.

$\frac{x - 4}{x^{2} + x + 1} \cdot \frac{x^{2} + x + 1}{(x - 4) (x - 1)}$

Step 7.3.2

Rewrite the expression.

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

Step 7.4

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

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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