Unit 11.3 Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions
In order to add or subtract fractions you need a common denominator. The same is true for adding and subtracting rational expressions
Finding a common denominator
1. What would you need to multiply [latex]\frac{x\ -\ 2\ }{(x\ -\ 3)}[/latex] by so that it has a common denominator with [latex]\frac{x\ -\ 2\ }{2(x\ -\ 3)}[/latex]?
Multiply both the numerator and denominator by 2
[latex]\frac{\ x\ -\ 2\ }{x\ -\ 3}\ \cdot\ \frac{2}{2}\ =\ \frac{2(x\ -\ 2)}{2(x\ -\ 3)}[/latex]
Now that the fractions have a common denominator, we would be able to add or subtract them
2. What would you need to multiply [latex]\frac{(x\ -\ 1)}{(x\ -\ 6)}[/latex] by so that is has a common denominator with [latex]\frac{x^2\ -\ 3}{(x\ +4)(x\ -\ 6)}[/latex]?
Multiply both the numerator and denominator by (x + 4)
[latex]\frac{x\ -\ 1}{(x\ -\ 6)}\ \cdot\frac{(x+4)}{(x+4)}=\frac{(x\ -\ 1)(x+4)\ }{(x\ -\ 6)(x+4)}[/latex]
Now that the fractions have a common denominator, we would be able to add or subtract them
Sometimes finding a common denominator requires factoring of the denominators so that it is easier to see what’s “missing” for the terms to be common. You can also find the LCM of the fractions. Once the common denominator is found, you can add or subtract the values in the numerator. Same with fractions, the common denominator will stay the same
Adding a Rational Expression
Add [latex]\frac{1}{x + 2}[/latex] and [latex]\frac{3}{x - 2}[/latex]
[latex]\frac{1}{x + 2} + \frac{3}{x - 2}[/latex]
Find the values that you need to multiply by to get a common denominator
The first fraction needs to be multiplied by (x – 2)
[latex]\frac{1}{x+2}\ \cdot\frac{(x\ -\ 2)}{(x\ -\ 2)}\ =\ \frac{x\ -\ 2}{(x+2)(x\ -\ 2)}[/latex]
The second fraction needs to be multiplied by (x+2)
[latex]\frac{3}{x\ -\ 2}\ \cdot\frac{(x\ +\ 2)}{(x\ +\ 2)}\ =\ \frac{3(x\ +\ 2)}{(x+2)(x\ -\ 2)}[/latex]
Rewrite the fractions
[latex]\frac{(x\ -\ 2)}{(x+2)(x\ -\ 2)}+\frac{3(x\ +\ 2)}{(x+2)(x\ -\ 2)}[/latex]
Now they have a common denominator, we can add the numerators
[latex]\frac{(x\ -\ 2)\ +\ 3(x+2)}{(x+2)(x\ -\ 2)}[/latex]
Simplify the numerator by expanding any multiplication
[latex]\frac{x\ -\ 2\ +\ 3x+6}{(x+2)(x\ -\ 2)}[/latex]
Combine like terms
[latex]\frac{\ 2x+4}{(x+2)(x\ -\ 2)}[/latex]
Refactor the numerator
[latex]\frac{4x\ +4}{(x+2)(x\ -\ 2)}[/latex]
Cancel out any like terms (if there are any). In this example, there are no more like terms to cancel out
Simplify
[latex]\frac{4(x\ +1)}{(x+2)(x\ -\ 2)}[/latex]