key term - Repeated Linear Factors

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.

5 Must Know Facts For Your Next Test

1. 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.
2. 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.
3. 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.
4. The coefficients of the partial fraction terms involving repeated linear factors are determined by solving a system of linear equations.
5. 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.

Related terms

Partial Fraction Decomposition:

The process of breaking down a rational expression into a sum of simpler rational expressions with linear or quadratic denominators.

Linear Factor: A factor of the form $(x-a)$, where $a$ is a constant.

Multiplicity:

The number of times a factor appears in the denominator of a rational expression.