Find the partial-fraction decomposition. (2 x^3+x^2-x-1)/(x^4+x^3)

Name: Problem 55, Chapter 8: Precalculus
Uploaded: 2020-06-16T14:49:48-08:00
Duration: 3 min 12 s
Channel: Thomas Emment
Description: Find the partial-fraction decomposition. (2 x^3+x^2-x-1)/(x^4+x^3)

step 1 Okay, we'd be given this partial fraction. Oh, we've given this fraction and we'd like to put it

Find the partial-fraction decomposition. (2...

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function (a ratio of two polynomials) as a sum of simpler fractions. This technique is particularly useful in integration, solving differential equations, and analyzing complex rational expressions by breaking them into parts that are easier to handle.

Rational Functions

A rational function is any function that can be written as the ratio of two polynomials. It is important to understand the properties of rational functions – especially their behavior at infinity and near any potential singularities – when applying techniques such as partial fraction decomposition.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of its irreducible factors. This is a key step in partial fraction decomposition because it allows one to identify the simpler components (usually linear or quadratic factors) over which the rational function can be split.

Equating Coefficients

Equating coefficients is an algebraic technique where the coefficients of like powers of the variable on both sides of an equation are set equal to each other. This approach is essential in partial fraction decomposition when determining the unknown constants in the numerators of the decomposed fractions.

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