How to add/simplify rational expression

Question

How do I add and simplify this rational expression step by step?? 

 

     5      +       5      

    x²            x²+x

Answer 1

You have two terms, each being a rational (although the division bars separating numerators from denominators are missing).

 

Factor each denominator: one is x² and the other is x(x+1).

 

By inspection, choose the least common denominator: LCD is x²(x+1).  Do you see how this contains each of the individual denominators in the problem?

 

Now, "pump up" the first individual denominator to make it agree with the LCD. This means that you need to multiply the first individual denominator by (x+1).  So you need to multiply that first numerator by (x+1) also.

 

Now pump up the second individual denom. by x to make it agree with the LCF. Therefore also multiply that original numerator of the second term by x.

 

You can now put 5(x+1) + 5x over x²(x+1) and simplify the numerator.

 

The answer is 5(2x+1) ALL OVER the LCD.

Answer 2

To start off, we can factor the denominators where needed.

 

5 / x² = 5 / [x(x + 1)]

 

 

We rewrite the expression using the lowest common denominator (LCD).  This is like adding fractions.  Remember doing this in elementary school?

 

The LCD is  x² * x(x + 1) = x³(x + 1)

 

Using the LCD, the new expression becomes

 

[5x(x + 1) + 5x²] / [x³(x + 1)]

 

 

Now we simplify the numerator.

 

(5x² + 5x + 5x²) / (x³(x + 1)) =

 

(10x² + 5x) / (x³(x + 1))

 

 

Now we can factor by taking out GCFs.

 

[5x(2x + 1)] / [x³(x + 1)] =

 

[5(2x + 1)] / [x²(x + 1)]

 

 

The expression can no longer be simplified.