Algebra Examples

$\frac{x + 1}{x^{2} - x - 20} - \frac{x + 4}{x^{2} - 4 x - 5} + \frac{x + 5}{x^{2} + 5 x + 4}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - x - 20$ using the AC method.

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Step 1.1.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

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

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Step 1.2.1

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

$- 5, 1$

Step 1.2.2

Write the factored form using these integers.

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

Step 1.3

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

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Step 1.3.1

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 $5$ .

$1, 4$

Step 1.3.2

Write the factored form using these integers.

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

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{x + 1}{(x - 5) (x + 4)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x + 1}{(x - 5) (x + 4)} \cdot \frac{x + 1}{x + 1} - \frac{x + 4}{(x - 5) (x + 1)} + \frac{x + 5}{(x + 1) (x + 4)}$

Step 2.2

Multiply $\frac{x + 1}{(x - 5) (x + 4)}$ by $\frac{x + 1}{x + 1}$ .

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

Step 2.3

Multiply $\frac{x + 4}{(x - 5) (x + 1)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(x + 1) (x + 1)}{(x - 5) (x + 4) (x + 1)} - (\frac{x + 4}{(x - 5) (x + 1)} \cdot \frac{x + 4}{x + 4}) + \frac{x + 5}{(x + 1) (x + 4)}$

Step 2.4

Multiply $\frac{x + 4}{(x - 5) (x + 1)}$ by $\frac{x + 4}{x + 4}$ .

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

Step 2.5

Multiply $\frac{x + 5}{(x + 1) (x + 4)}$ by $\frac{x - 5}{x - 5}$ .

$\frac{(x + 1) (x + 1)}{(x - 5) (x + 4) (x + 1)} - \frac{(x + 4) (x + 4)}{(x - 5) (x + 1) (x + 4)} + \frac{x + 5}{(x + 1) (x + 4)} \cdot \frac{x - 5}{x - 5}$

Step 2.6

Multiply $\frac{x + 5}{(x + 1) (x + 4)}$ by $\frac{x - 5}{x - 5}$ .

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

Step 2.7

Reorder the factors of $(x - 5) (x + 1) (x + 4)$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + 1 (x + 1) - (x + 4) (x + 4) + (x + 5) (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - (x + 4) (x + 4) + (x + 5) (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $x$ by $x$ .

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

Step 3.2.2.1.2

Multiply $x$ by $1$ .

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

Step 3.2.2.1.3

Multiply $x$ by $1$ .

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

Step 3.2.2.1.4

Multiply $1$ by $1$ .

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

Step 3.2.2.2

Add $x$ and $x$ .

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

Step 3.2.3

Apply the distributive property.

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

Step 3.2.4

Multiply $- 1$ by $4$ .

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

Step 3.2.5

Expand $(- x - 4) (x + 4)$ using the FOIL Method.

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Step 3.2.5.1

Apply the distributive property.

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

Step 3.2.5.2

Apply the distributive property.

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

Step 3.2.5.3

Apply the distributive property.

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

Step 3.2.6

Simplify and combine like terms.

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Step 3.2.6.1

Simplify each term.

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Step 3.2.6.1.1

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

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Step 3.2.6.1.1.1

Move $x$ .

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

Step 3.2.6.1.1.2

Multiply $x$ by $x$ .

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

Step 3.2.6.1.2

Multiply $4$ by $- 1$ .

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

Step 3.2.6.1.3

Multiply $- 4$ by $4$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 4 x - 4 x - 16 + (x + 5) (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.6.2

Subtract $4 x$ from $- 4 x$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + (x + 5) (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.7

Expand $(x + 5) (x - 5)$ using the FOIL Method.

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Step 3.2.7.1

Apply the distributive property.

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x (x - 5) + 5 (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.7.2

Apply the distributive property.

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x \cdot x + x \cdot - 5 + 5 (x - 5)}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.7.3

Apply the distributive property.

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.8

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

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Step 3.2.8.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x \cdot x - 5 x + 5 x + 5 \cdot - 5}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.8.2

Add $- 5 x$ and $5 x$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x \cdot x + 0 + 5 \cdot - 5}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.8.3

Add $x \cdot x$ and $0$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x \cdot x + 5 \cdot - 5}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.9

Simplify each term.

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Step 3.2.9.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x^{2} + 5 \cdot - 5}{(x + 1) (x + 4) (x - 5)}$

Step 3.2.9.2

Multiply $5$ by $- 5$ .

$\frac{x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x^{2} - 25}{(x + 1) (x + 4) (x - 5)}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} + 2 x + 1 - x^{2} - 8 x - 16 + x^{2} - 25$ .

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Step 3.3.1.1

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 x + 1 + 0 - 8 x - 16 + x^{2} - 25}{(x + 1) (x + 4) (x - 5)}$

Step 3.3.1.2

Add $2 x + 1$ and $0$ .

$\frac{2 x + 1 - 8 x - 16 + x^{2} - 25}{(x + 1) (x + 4) (x - 5)}$

Step 3.3.2

Subtract $8 x$ from $2 x$ .

$\frac{- 6 x + 1 - 16 + x^{2} - 25}{(x + 1) (x + 4) (x - 5)}$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Subtract $16$ from $1$ .

$\frac{- 6 x - 15 + x^{2} - 25}{(x + 1) (x + 4) (x - 5)}$

Step 3.3.3.2

Subtract $25$ from $- 15$ .

$\frac{- 6 x + x^{2} - 40}{(x + 1) (x + 4) (x - 5)}$

Step 3.3.3.3

Reorder $- 6 x$ and $x^{2}$ .

$\frac{x^{2} - 6 x - 40}{(x + 1) (x + 4) (x - 5)}$

Step 4

Factor $x^{2} - 6 x - 40$ using the AC method.

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Step 4.1

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 $- 40$ and whose sum is $- 6$ .

$- 10, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 10) (x + 4)}{(x + 1) (x + 4) (x - 5)}$

Step 5

Cancel the common factor of $x + 4$ .

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Step 5.1

Cancel the common factor.

$\frac{(x - 10) (x + 4)}{(x + 1) (x + 4) (x - 5)}$

Step 5.2

Rewrite the expression.

$\frac{x - 10}{(x + 1) (x - 5)}$