Repeated Linear Factors

Definition

Repeated linear factors are factors in a polynomial that occur more than once, indicating that the corresponding root has a multiplicity greater than one. These factors play a significant role in partial fraction decomposition, as they require special treatment to express rational functions in a sum of simpler fractions. Understanding how to handle these repeated factors is essential for finding the proper form of the decomposition and ensures that all aspects of the function are represented accurately.

5 Must Know Facts For Your Next Test

1. In partial fraction decomposition, when a linear factor is repeated, it must be expressed in multiple terms, each raised to successive powers up to the multiplicity of the factor.
2. If a linear factor appears with multiplicity 'm', the corresponding terms in the decomposition will include fractions for each power from 1 to 'm'.
3. For example, if (x - a) is a repeated linear factor of multiplicity 3, the decomposition will include terms like A/(x - a) + B/(x - a)^2 + C/(x - a)^3.
4. To find the coefficients for these terms, you typically multiply through by the common denominator and equate coefficients after substituting appropriate values for x.
5. Careful attention must be paid to repeated factors during integration and limits, as their presence can lead to different behavior compared to simple linear factors.

Review Questions

• How does one identify and express repeated linear factors when performing partial fraction decomposition?
  • To identify repeated linear factors in a polynomial, look for any factors that appear more than once in its factored form. When expressing these factors in partial fraction decomposition, each factor must be included in terms for every multiplicity it has. For example, if (x - a) appears twice, your decomposition would include A/(x - a) and B/(x - a)^2. This ensures that all contributions of that factor are represented.
• Explain how you would find the coefficients of the partial fraction decomposition involving repeated linear factors.
  • To find the coefficients when dealing with repeated linear factors, you start by multiplying the entire equation by the common denominator to eliminate the fractions. Then, you substitute convenient values for x that simplify calculations or allow you to isolate terms. If needed, differentiate or use algebraic methods to equate coefficients for terms at each power of the repeated factor, enabling you to solve for each unknown coefficient effectively.
• Analyze the impact of neglecting repeated linear factors on solving integrals involving rational functions.
  • Neglecting repeated linear factors when integrating rational functions can lead to incorrect results because it fails to account for the full contribution of those factors. Each power of a repeated factor influences the behavior of the function near its roots. If these are not included in the decomposition properly, it can result in missing necessary logarithmic terms during integration. Thus, acknowledging their presence ensures accurate computation and meaningful results.

Related terms

Polynomial: An algebraic expression consisting of variables raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

Partial Fraction Decomposition: The process of breaking down a complex rational function into a sum of simpler fractions, which makes integration and other operations easier.

Multiplicity: The number of times a particular root appears in the factorization of a polynomial, indicating how many times that root contributes to the polynomial's behavior.