Addition is a fundamental mathematical operation that combines two or more quantities to find their sum. It is one of the basic arithmetic operations, along with subtraction, multiplication, and division, and is essential in the context of rational expressions.
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Addition of rational expressions is performed by finding a common denominator and then adding the numerators.
The common denominator is the least common multiple (LCM) of the denominators of the rational expressions being added.
Addition of rational expressions can be used to simplify complex fractions and expressions involving variables.
The properties of addition, such as commutativity and associativity, hold true for the addition of rational expressions.
Addition of rational expressions is a crucial skill in solving equations and inequalities involving rational functions.
Review Questions
Explain the process of adding two rational expressions with different denominators.
To add two rational expressions with different denominators, you first need to find the least common multiple (LCM) of the denominators. This will be the common denominator for the addition. Then, you rewrite each rational expression with the common denominator by multiplying the numerator and denominator by the appropriate factors. Finally, you add the numerators and write the result over the common denominator.
Describe how the properties of addition, such as commutativity and associativity, apply to the addition of rational expressions.
The properties of addition, including commutativity (a + b = b + a) and associativity ((a + b) + c = a + (b + c)), hold true for the addition of rational expressions. This means that the order in which rational expressions are added does not affect the final result, and expressions can be grouped in different ways without changing the sum. These properties are important when simplifying complex rational expressions involving multiple additions.
Analyze the role of addition of rational expressions in solving equations and inequalities involving rational functions.
Addition of rational expressions is a crucial skill in solving equations and inequalities involving rational functions. When dealing with rational functions, it is often necessary to manipulate the expressions by adding them together to isolate the variable of interest. This allows you to solve for the unknown value or determine the range of values that satisfy the given inequality. The ability to accurately add rational expressions is essential for successfully solving problems involving rational functions.
A polynomial is an expression that consists of variables and coefficients, where the variables are only raised to non-negative integer powers.
Algebraic Operations: Algebraic operations are the fundamental mathematical manipulations performed on algebraic expressions, such as addition, subtraction, multiplication, and division.