Find the partial fraction decomposition for each rational expression. (-3 x-5)/((x+3)(x-1))

Name: Problem 12, Chapter 8: Precalculus Functions and Graphs
Uploaded: 2020-04-16T23:02:04-08:00
Duration: 1 min 20 s
Channel: Xiaomeng Zhang
Description: Find the partial fraction decomposition for each rational expression. (-3 x-5)/((x+3)(x-1))

step 1 So for this one, we have minus 3x minus 5 over x plus 3 times x minus 1 equals to A over x plus

Find the partial fraction decomposition for each rational...

Key Concepts

Solving Systems of Equations

Solving the system of equations obtained from equating coefficients allows you to determine the values of the unknown constants in the partial fractions. This step often involves basic algebraic methods such as substitution or elimination.

Equating Coefficients

Once the rational expression is represented as a sum of partial fractions, the next step commonly involves equating the coefficients of like terms in the numerators of the expressions when their denominators are made common. This provides a system of equations that can be solved to find the unknown constants.

Factorization of the Denominator

The factorization of the denominator involves expressing the denominator polynomial as a product of its irreducible factors. This step is critical in partial fraction decomposition because the simpler fractions are built around these factors.

Rational Expressions

A rational expression is a ratio of two polynomials. Understanding the structure of rational expressions, including the degree of the numerator and denominator, is crucial for applying techniques like partial fraction decomposition correctly.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational expression into a sum of simpler rational expressions whose denominators are typically factors of the original denominator. This method simplifies many processes in algebra and calculus, such as integration and solving differential equations.

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