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

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

Step 1

Simplify the denominator.

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

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

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 $b = 1$ .

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

Step 2

To write $\frac{3 x + 1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(3 x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $3 x$ and $x$ .

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

Add $4 x$ and $x$ .

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