Find the partial fraction decomposition. (4 x^2-15 x-1)/((x-1)(x+2)(x-3))

Name: Problem 5, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-06T20:33:26-08:00
Duration: 1 min 23 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (4 x^2-15 x-1)/((x-1)(x+2)(x-3))

step 1 Here we're given a fraction that we have to decompose into different partial fractions, right? S

Find the partial fraction decomposition. (4 x^2-15...

Key Concepts

Equating Coefficients

Equating coefficients is a method used after setting up the partial fractions. By expanding and combining like terms, one can compare the coefficients of corresponding powers of the variable on both sides of the equation. This process generates a system of linear equations which, when solved, yields the unknown constants in the partial fractions.

Factorization of Polynomials

Factorization of polynomials involves expressing the polynomial as a product of its irreducible factors. This step is essential in partial fraction decomposition because the denominator must be factored completely to identify the simpler fractions that will sum to the original rational function.

Systems of Linear Equations

In the context of partial fraction decomposition, systems of linear equations arise from equating coefficients. Solving these systems is essential to find the values of the unknown constants in the decomposition. Various methods, such as substitution, elimination, or matrix techniques, can be used to solve these systems efficiently.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into a sum of simpler fractions whose denominators are factors of the original denominator. This process simplifies the integration, differentiation, or algebraic manipulation of rational functions by reducing them to sums of basic fractions with linear or quadratic factors.

Rational Functions

A rational function is a ratio of two polynomials. Understanding the structure of a rational function is crucial for decomposition because the degrees of the numerator and denominator determine if polynomial division is needed before applying the partial fraction method. It also guides the selection of appropriate factors for the decomposition process.

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