Repeated Linear Factors
from class:
College Algebra
Definition
Repeated linear factors refer to the occurrence of linear factors, such as $(x-a)$, that appear more than once in the denominator of a rational expression. These repeated factors are crucial in the context of partial fraction decomposition, as they require a specific approach to properly handle them.
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5 Must Know Facts For Your Next Test
- Repeated linear factors in the denominator of a rational expression indicate that the partial fraction decomposition will involve a sum of terms with different forms, depending on the multiplicity of the factor.
- When a linear factor appears multiple times in the denominator, the corresponding partial fraction terms will include a sum of fractions with different powers of the linear factor in the denominator.
- The number of partial fraction terms corresponding to a repeated linear factor is equal to its multiplicity, the number of times it appears in the denominator.
- The coefficients of the partial fraction terms involving repeated linear factors are determined by solving a system of linear equations.
- Properly handling repeated linear factors is crucial for successfully performing partial fraction decomposition and evaluating integrals involving rational expressions.
Review Questions
- Explain the significance of repeated linear factors in the context of partial fraction decomposition.
- Repeated linear factors in the denominator of a rational expression indicate that the partial fraction decomposition will involve a sum of terms with different forms, depending on the multiplicity of the factor. The number of partial fraction terms corresponding to a repeated linear factor is equal to its multiplicity, and the coefficients of these terms are determined by solving a system of linear equations. Properly handling repeated linear factors is crucial for successfully performing partial fraction decomposition and evaluating integrals involving rational expressions.
- Describe the general form of the partial fraction terms corresponding to a repeated linear factor.
- When a linear factor appears multiple times in the denominator of a rational expression, the corresponding partial fraction terms will include a sum of fractions with different powers of the linear factor in the denominator. For example, if a linear factor $(x-a)$ appears $n$ times in the denominator, the partial fraction terms will have the form $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$, where the coefficients $A_1, A_2, \dots, A_n$ are determined by solving a system of linear equations.
- Analyze the relationship between the multiplicity of a repeated linear factor and the number of partial fraction terms required to represent it.
- The number of partial fraction terms corresponding to a repeated linear factor is equal to its multiplicity, the number of times it appears in the denominator. This means that if a linear factor $(x-a)$ appears $n$ times in the denominator of a rational expression, the partial fraction decomposition will require $n$ terms of the form $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$, where the coefficients $A_1, A_2, \dots, A_n$ are determined by solving a system of linear equations. The multiplicity of the repeated linear factor directly determines the complexity of the partial fraction decomposition process.
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