Find the partial-fraction decomposition for each rational function. (x^3)/((x^2+9)^2)

Name: Problem 31, Chapter 8: Precalculus
Uploaded: 2020-06-16T11:27:45-08:00
Duration: 2 min 40 s
Channel: Thomas Emment
Description: Find the partial-fraction decomposition for each rational function. (x^3)/((x^2+9)^2)

step 1 Okay, we've been given this fraction x cubed over x squared plus 9 squared. So I want to put it

Find the partial-fraction decomposition for each rational...

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials. In many areas of mathematics, especially in integration, it is useful to express a rational function in a simpler form by decomposing it into a sum of fractions which individually have simpler denominators.

Partial Fractions Decomposition

Partial fractions decomposition is a method used to break down a complex rational expression into a sum of simpler fractions. This technique is particularly useful for integration and solving differential equations, as it transforms a complicated expression into a form where standard methods can be applied.

Irreducible Quadratic Factors

When a denominator contains quadratic factors that cannot be factored further over the real numbers, they are termed irreducible. In partial fractions decomposition, these factors require numerators that are linear polynomials, ensuring that all possible degrees are properly accounted for in the decomposition.

Repeated Factors

Repeated factors occur when a factor in the denominator is raised to a power greater than one. Each occurrence of the repeated factor must be represented separately in the decomposition, typically with terms corresponding to each power from one up to the highest power present. This approach is essential for accurately representing the original rational function.

Method of Equating Coefficients

The method of equating coefficients is a common technique for finding the unknown constants in a partial fractions decomposition. After setting up the assumed form of the decomposition, both sides of the equation are expanded and the coefficients of like powers of the variable are set equal to each other. This yields a system of equations to solve for the unknown constants.

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