Find the partial fraction decomposition. (2 x+3)/((x-1)^2) |...

Name: Problem 9, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T01:21:16-08:00
Duration: 2 min 39 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (2 x+3)/((x-1)^2)

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Key Concepts

Partial Fraction Decomposition

This is a method used to express a complex rational function as a sum of simpler fractions whose denominators are typically factors of the original denominator. It simplifies many operations in calculus, such as integration and solving differential equations, by breaking down functions into manageable parts.

Rational Functions

A rational function is a quotient of two polynomials. Understanding its structure is key to applying partial fraction decomposition because the factors of the denominator determine the form of the decomposed fractions.

Repeated Linear Factors

When the denominator of a rational function contains repeated linear factors, each repeated factor contributes multiple terms to the decomposition (one for each power up to the multiplicity). This is crucial for setting up the correct form of the partial fractions to be solved.

Systems of Linear Equations

After setting up the decomposition’s structure, finding the unknown coefficients typically requires equating coefficients of like powers of the variable, which results in a system of linear equations. Solving this system yields the values needed to complete the decomposition.

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