Simplify (3x)/(x+3)+16/(x-3)-54/(x^2-9)

$\frac{3 x}{x + 3} + \frac{16}{x - 3} - \frac{54}{x^{2} - 9}$

Step 1

Simplify the denominator.

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

$\frac{3 x}{x + 3} + \frac{16}{x - 3} - \frac{54}{x^{2} - 3^{2}}$

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 $b = 3$ .

$\frac{3 x}{x + 3} + \frac{16}{x - 3} - \frac{54}{(x + 3) (x - 3)}$

Step 2

To write $\frac{3 x}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3 x}{x + 3} \cdot \frac{x - 3}{x - 3} + \frac{16}{x - 3} - \frac{54}{(x + 3) (x - 3)}$

Step 3

To write $\frac{16}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{3 x}{x + 3} \cdot \frac{x - 3}{x - 3} + \frac{16}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{54}{(x + 3) (x - 3)}$

Step 4

Write each expression with a common denominator of $(x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{3 x (x - 3)}{(x + 3) (x - 3)} + \frac{16}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{54}{(x + 3) (x - 3)}$

Multiply $\frac{16}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$\frac{3 x (x - 3)}{(x + 3) (x - 3)} + \frac{16 (x + 3)}{(x - 3) (x + 3)} - \frac{54}{(x + 3) (x - 3)}$

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

$\frac{3 x (x - 3)}{(x + 3) (x - 3)} + \frac{16 (x + 3)}{(x + 3) (x - 3)} - \frac{54}{(x + 3) (x - 3)}$

Step 5

Simplify terms.

Combine the numerators over the common denominator.

$\frac{3 x (x - 3) + 16 (x + 3)}{(x + 3) (x - 3)} - \frac{54}{(x + 3) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{3 x (x - 3) + 16 (x + 3) - 54}{(x + 3) (x - 3)}$

Step 6

Simplify each term.

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 3 + 16 (x + 3) - 54}{(x + 3) (x - 3)}$

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

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot - 3 + 16 (x + 3) - 54}{(x + 3) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot - 3 + 16 (x + 3) - 54}{(x + 3) (x - 3)}$

Multiply $- 3$ by $3$ .

$\frac{3 x^{2} - 9 x + 16 (x + 3) - 54}{(x + 3) (x - 3)}$

Apply the distributive property.

$\frac{3 x^{2} - 9 x + 16 x + 16 \cdot 3 - 54}{(x + 3) (x - 3)}$

Multiply $16$ by $3$ .

$\frac{3 x^{2} - 9 x + 16 x + 48 - 54}{(x + 3) (x - 3)}$

Step 7

Simplify by adding terms.

Add $- 9 x$ and $16 x$ .

$\frac{3 x^{2} + 7 x + 48 - 54}{(x + 3) (x - 3)}$

Subtract $54$ from $48$ .

$\frac{3 x^{2} + 7 x - 6}{(x + 3) (x - 3)}$

Step 8

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$\frac{3 x^{2} + 7 (x) - 6}{(x + 3) (x - 3)}$

Rewrite $7$ as $- 2$ plus $9$

$\frac{3 x^{2} + (- 2 + 9) x - 6}{(x + 3) (x - 3)}$

Apply the distributive property.

$\frac{3 x^{2} - 2 x + 9 x - 6}{(x + 3) (x - 3)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} - 2 x) + 9 x - 6}{(x + 3) (x - 3)}$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 9

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 x - 2}{x - 3}$

Simplify (3x)/(x+3)+16/(x-3)-54/(x^2-9)

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