What Are Rational Expressions?
A rational expression is the ratio of two polynomials, as shown below.
Both P(x) and Q(x) are polynomials. Of course Q(x) cannot equal zero.
Let’s look at the simplest rational expression.
Let P(x) = 1 and Q(x) = x.
The rational expression becomes:
Does this expression remind you of a fraction? You can use the properties of fractions to combine rational expressions.
Let’s look at an example.
If we treat each of these as a fraction, can you see that these are two “fractions” with the same denominator.
Combine rational expressions using the properties of fractions. Now let’s look at an example of two rational numbers with different denominators.
In order to combine these two expressions, they need a common denominator. Notice how one denominator is a multiple of the other. Here’s how to find a common denominator:
Here are some examples of combining rational expressions using the four basic operations.
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Summary
A rational expression is a mathematical expression that represents the quotient, or ratio, of two polynomial expressions. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. A polynomial is an algebraic expression consisting of variables, coefficients, and exponents, with addition, subtraction, and multiplication operations but not division by variables.
Here's the general form of a rational expression:
Where:
- P(x) is the numerator, and it's a polynomial expression.
- Q(x) is the denominator, and it's also a polynomial expression.
- x is the variable.
Rational expressions can have various forms and complexities. They can contain constants, variables, and exponents, and you can perform operations like addition, subtraction, multiplication, and division with them. When working with rational expressions, you may need to simplify them, find their domain (the values of x for which the expression is defined), and solve equations involving them. They often arise in mathematics, especially in algebra, calculus, and other areas of mathematics that involve functions and equations.
