A/(x-a) is a type of rational expression, where A represents a constant and a represents a value that the variable x is subtracted from. This expression is commonly encountered in the context of partial fractions, a technique used to decompose a rational expression into a sum of simpler rational expressions.
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The expression A/(x-a) represents a linear factor in the denominator of a rational expression.
When dealing with partial fractions, the expression A/(x-a) is used to handle linear factors in the denominator.
The value of 'a' in the expression A/(x-a) represents a root or zero of the denominator polynomial.
The constant 'A' in the expression A/(x-a) is determined through the process of partial fraction decomposition.
Integrating or finding the antiderivative of A/(x-a) is a common application of the partial fractions technique.
Review Questions
Explain how the expression A/(x-a) is used in the context of partial fractions.
In the context of partial fractions, the expression A/(x-a) is used to handle linear factors in the denominator of a rational expression. The value of 'a' represents a root or zero of the denominator polynomial, and the constant 'A' is determined through the process of partial fraction decomposition. This allows the original rational expression to be broken down into a sum of simpler rational expressions, which can be more easily integrated or manipulated.
Describe the role of the constant 'A' in the expression A/(x-a) within the partial fractions technique.
The constant 'A' in the expression A/(x-a) is a crucial component in the partial fractions technique. The value of 'A' is determined through the process of partial fraction decomposition, where the original rational expression is broken down into a sum of simpler rational expressions. The constant 'A' represents the coefficient or weight of the linear factor (x-a) in the denominator of the original rational expression. Knowing the value of 'A' is essential for successfully integrating or manipulating the resulting partial fraction components.
Analyze how the expression A/(x-a) is related to the roots or zeros of the denominator polynomial in a rational expression.
The expression A/(x-a) is directly related to the roots or zeros of the denominator polynomial in a rational expression. The value of 'a' in the expression represents a root or zero of the denominator polynomial, where the factor (x-a) becomes zero. This linear factor in the denominator is then handled using the partial fractions technique, with the constant 'A' determined through the decomposition process. Understanding the connection between the expression A/(x-a) and the roots or zeros of the denominator is crucial for successfully applying the partial fractions method to solve problems involving rational expressions.
Related terms
Partial Fractions: A method used to express a rational expression as a sum of simpler rational expressions, which can be more easily integrated or manipulated.
Decomposition: The process of breaking down a complex rational expression into a sum of simpler rational expressions, often using the method of partial fractions.