As we know, a rational number is one that can be expressed as a fraction. That is to say,
where a and b are integers and .
A rational expression, which is also called an algebraic fraction, can be expressed as a quotient of polynomials. That is to say,
where a and b are polynomials and .
We can add and subtract rational expressions easily, but there are a few steps that we must take when the expressions have unlike denominators. First, let's look at adding a rational expression with like denominators.
Example 1
Add the following rational expressions:
To add rational expressions with like denominators, simply add the numerators.
So
There are a few steps you must take with rational expressions that have unlike denominators when you add or subtract them. They are as follows:
Example 2
Add .
Since the denominators are not the same, we must first find the least common denominator.
Since 4a and 5b have no common factors, the LCM is simply their product: .
That is to say, the LCD of the fractions is 20ab.
Rewrite the fractions using the LCD.
Example 3
Add .
Since the denominators are not the same, first we must find the LCD.
Here, the greatest common factor (GCF) of and is . So to find the LCM, we must divide the product by .
Next, we rewrite the fractions using the LCD.
Finally, we simplify by performing the calculations.
Example 4
Subtract
Since the denominators are not the same, we must find the LCD.
The LCM of and is .
That is to say, the LCD of and is .
So we will rewrite the fractions using the LCD.
Simplify the numerator on the first fraction.
Next, we subtract the numerators.
Then we simplify the numerator.
Adding and Subtracting Rational Expressions with Like Denominators
Simplifying Rational Expressions
College Algebra Diagnostic Tests
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