Simplify (2x+5)/(x^2-3x)-(3x+5)/(x^3-9x)-(x+1)/(x^2-9)

$\frac{2 x + 5}{x^{2} - 3 x} - \frac{3 x + 5}{x^{3} - 9 x} - \frac{x + 1}{x^{2} - 9}$

Step 1

Simplify each term.

Factor $x$ out of $x^{2} - 3 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{2 x + 5}{x \cdot x - 3 x} - \frac{3 x + 5}{x^{3} - 9 x} - \frac{x + 1}{x^{2} - 9}$

Factor $x$ out of $- 3 x$ .

$\frac{2 x + 5}{x \cdot x + x \cdot - 3} - \frac{3 x + 5}{x^{3} - 9 x} - \frac{x + 1}{x^{2} - 9}$

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x^{3} - 9 x} - \frac{x + 1}{x^{2} - 9}$

Simplify the denominator.

Factor $x$ out of $x^{3} - 9 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x \cdot x^{2} - 9 x} - \frac{x + 1}{x^{2} - 9}$

Factor $x$ out of $- 9 x$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x \cdot x^{2} + x \cdot - 9} - \frac{x + 1}{x^{2} - 9}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x (x^{2} - 9)} - \frac{x + 1}{x^{2} - 9}$

Rewrite $9$ as $3^{2}$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x (x^{2} - 3^{2})} - \frac{x + 1}{x^{2} - 9}$

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 $b = 3$ .

$\frac{2 x + 5}{x (x - 3)} - \frac{3 x + 5}{x (x + 3) (x - 3)} - \frac{x + 1}{x^{2} - 9}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 2

Find the common denominator.

Multiply $\frac{2 x + 5}{x (x - 3)}$ by $\frac{x + 3}{x + 3}$ .

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

Multiply $\frac{2 x + 5}{x (x - 3)}$ by $\frac{x + 3}{x + 3}$ .

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

Multiply $\frac{x + 1}{(x + 3) (x - 3)}$ by $\frac{x}{x}$ .

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

Multiply $\frac{x + 1}{(x + 3) (x - 3)}$ by $\frac{x}{x}$ .

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

Reorder the factors of $(x + 3) (x - 3) x$ .

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

Step 3

Simplify terms.

Combine the numerators over the common denominator.

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

Simplify each term.

Expand $(2 x + 5) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $2$ .

$\frac{2 x^{2} + 6 x + 5 x + 5 \cdot 3 - (x + 1) x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Multiply $5$ by $3$ .

$\frac{2 x^{2} + 6 x + 5 x + 15 - (x + 1) x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Add $6 x$ and $5 x$ .

$\frac{2 x^{2} + 11 x + 15 - (x + 1) x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{2 x^{2} + 11 x + 15 + (- x - 1 \cdot 1) x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} + 11 x + 15 + (- x - 1) x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{2 x^{2} + 11 x + 15 - x \cdot x - 1 x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

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

Move $x$ .

$\frac{2 x^{2} + 11 x + 15 - (x \cdot x) - 1 x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 11 x + 15 - x^{2} - 1 x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} + 11 x + 15 - x^{2} - x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Simplify terms.

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

$\frac{x^{2} + 11 x + 15 - x}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Subtract $x$ from $11 x$ .

$\frac{x^{2} + 10 x + 15}{x (x + 3) (x - 3)} + \frac{- (3 x + 5)}{x (x + 3) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{x^{2} + 10 x + 15 - (3 x + 5)}{x (x + 3) (x - 3)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{x^{2} + 10 x + 15 - (3 x) - 1 \cdot 5}{x (x + 3) (x - 3)}$

Multiply $3$ by $- 1$ .

$\frac{x^{2} + 10 x + 15 - 3 x - 1 \cdot 5}{x (x + 3) (x - 3)}$

Multiply $- 1$ by $5$ .

$\frac{x^{2} + 10 x + 15 - 3 x - 5}{x (x + 3) (x - 3)}$

Subtract $3 x$ from $10 x$ .

$\frac{x^{2} + 7 x + 15 - 5}{x (x + 3) (x - 3)}$

Subtract $5$ from $15$ .

$\frac{x^{2} + 7 x + 10}{x (x + 3) (x - 3)}$

Factor $x^{2} + 7 x + 10$ using the AC method.

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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

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

Simplify (2x+5)/(x^2-3x)-(3x+5)/(x^3-9x)-(x+1)/(x^2-9)

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