A rational expression is a mathematical expression that represents the ratio of two polynomials. It is a type of algebraic expression that can be simplified, multiplied, divided, added, or subtracted using specific rules and operations.
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Rational expressions can be divided by dividing the numerators and multiplying the denominators.
Multiplying rational expressions involves multiplying the numerators and multiplying the denominators.
Adding or subtracting rational expressions with a common denominator requires finding a common denominator and then adding or subtracting the numerators.
Complex rational expressions are rational expressions that have a rational expression in the numerator or denominator, or both.
Simplifying complex rational expressions involves applying the rules for simplifying rational expressions to the numerator and denominator separately.
Review Questions
Explain how to divide monomials in the context of rational expressions.
When dividing monomials in the context of rational expressions, the key is to recognize that a monomial is a special case of a polynomial, where the polynomial has only one term. To divide monomials in a rational expression, you can apply the rules for dividing polynomials by dividing the coefficients and subtracting the exponents of the variables in the numerator and denominator.
Describe the process of simplifying rational expressions.
Simplifying a rational expression involves reducing the expression to its simplest form by factoring the numerator and denominator and then canceling any common factors. This can be done by identifying the greatest common factor (GCF) of the numerator and denominator, and then dividing both the numerator and denominator by the GCF. The resulting expression will be an equivalent rational expression that is in its simplest form.
Analyze the steps required to add or subtract rational expressions with a common denominator.
To add or subtract rational expressions with a common denominator, the first step is to find the least common multiple (LCM) of the denominators. This will be the new common denominator. Then, you can rewrite each rational expression with the new common denominator by multiplying the numerator and denominator by the appropriate factors. Finally, you can add or subtract the numerators and keep the common denominator. This process ensures that the resulting expression is also a rational expression.
A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.
Simplify: The process of reducing a rational expression to its simplest form by factoring the numerator and denominator and canceling common factors.
Equivalent Rational Expressions: Two rational expressions that represent the same value, even though they may have different numerators and denominators.