# Rational Expressions - Add/Subtract - Part 2

## Rational Expressions - Add/Subtract - Part 2- How it Works - Video

### Example 1

Example 1:

Here we have the expression (x2 - 5x + 4) / (x2 - 2x - 8) - (x2 - 9) / (x2 + 5x + 6). The best bet when dealing with rational expressions is to factor them and see if we can cancel any factors out so we can make it smaller.

Factor: x2 - 2x - 8

Factor: x2 - 2x - 8

Factor: x2 - 9

Factor: x2 + 5x + 6

So x2 - 5x + 4 becomes (x - 1)(x - 4).

So x2 - 2x - 8 becomes (x + 2)(x - 4).

So x2 - 9 becomes (x - 3)(x + 3). Remember this is a special case because it is a difference of squares.

So x2 + 5x + 6 becomes (x + 2)(x + 3).

(x2 - 5x + 4) / (x2 - 2x - 8)      -      (x2 - 9) / (x2 + 5x + 6)

(x - 1)(x - 4) / (x + 2)(x - 4)      -      (x - 3)(x + 3) / (x + 2)(x + 3)

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

[ (x - 1) - (x - 3) ]    /    (x + 2)

[ x - 1 - x + 3 ]    /    (x + 2)

x     /     (x + 2)

Given

Factored each polynomial

Canceled out the factors (x - 4) on the left and (x + 3) on the right

Combined fractions

Distributed the negative in front of the parenthesis

Combined like terms

### Example 2

Example 2:

Here we have the expression (x2 + 5x) / (2x2 + 6x - 20) - (x2 - 1) / (x2 - 3x + 2). The best bet when dealing with rational expressions is to factor them and see if we can cancel any factors out so we can make it smaller.

Factor: x2 + 5x

Factor: 2x2 + 6x - 20

Factor: x2 - 1

Factor: x2 - 3x + 2

So x2 + 5x becomes x(x +5)

So 2x2 + 6x + 20 becomes 2(x - 2)(x + 5).

So x2 - 1 becomes (x - 1)(x + 1). Remember this is a special case because it is a difference of squares.

So x2 - 3x + 2 becomes (x - 1)(x - 2).

(x2 + 5x) / (2x2 + 6x - 20)      +      (x2 - 1) / (x2 - 3x + 2)

x(x + 5) / [ 2(x - 2)(x + 5) ]      +      (x - 1)(x + 1) / (x - 1)(x -2)

x / [ 2(x - 2) ]     +      (x + 1) / (x - 2)

x / [ 2(x - 2) ]     +      (x + 1) / (x - 2)    *   2/2

x / [ 2(x - 2) ]     +      2(x + 1) / [ 2(x - 2) ]

{ x + 2(x + 1) }      /      [ 2(x - 2) ]

{ x + 2x + 2) }      /      [ 2(x - 2) ]

{ 3x + 2 }      /      [ 2(x - 2) ]

Given

Factored each polynomial

Canceled out the factors (x + 5) on the left and (x - 1) on the right

Multiplied the fraction on the right by 2/2

Smashed the numbers together to get of multiplication sign

Combined fractions

Distributed the negative in front of the parenthesis

Combined like terms