Question

Subtract rational expression. Show work please.

Answer 1

Given:

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

To find:

The simplified rational expression by subtraction.

Solution:

Let us factor $x^2-1$. It can be written as $x^2-1^2$.

$x^2-1^2=(x-1)(x+1)$ using algebraic identity.

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

LCM of $x-1,(x+1)(x-1)=(x+1)(x-1)$

Make the denominators same using LCM.

Multiply and divide the first term by (x + 1) to make the denominator same.

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

Now, denominators are same, you can subtract the fractions.

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

Expand $(x+4)(x+1)-5$.

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

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

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