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

Name: Problem 31, Chapter 10: Precalculus Mathematics for Calculus
Uploaded: 2020-06-10T11:03:51-08:00
Duration: 4 min 33 s
Channel: Thomas Emment
Description: Find the partial fraction decomposition of the rational function. (4 x^2-x-2)/(x^4+2 x^3)

Find the partial fraction decomposition of the rational...

Key Concepts

Repeated Linear Factors

When the denominator contains linear factors that repeat, each repeated factor must be represented in the decomposition by a series of terms with incrementally increasing powers in the denominator. This ensures that the decomposition fully represents the original function.

Partial Fraction Decomposition

This is a method for expressing a complex rational function as a sum of simpler rational functions, often with denominators that are factors of the original denominator. It simplifies arithmetic operations, integration, and solving differential equations by breaking a complicated expression into more manageable pieces.

Polynomial Factorization

Before decomposing a rational function into partial fractions, it is necessary to factor the denominator (and sometimes the numerator) into irreducible polynomials. This process reveals the structure of the factors, determines whether there are repeated factors, and guides the form of the partial fraction decomposition.

Proper Rational Function

A rational function is called proper if the degree of its numerator is less than the degree of its denominator. For partial fraction decomposition to be directly applied, the rational function must be proper. If it is not, polynomial long division is performed first to convert it into a proper function plus a polynomial.

Method of Equating Coefficients

After setting up the partial fraction form with unknown constants, the method of equating coefficients involves expanding the expression and matching the coefficients of corresponding powers of the variable on both sides of the equation. This algebraic technique is used to solve for the unknown constants in the decomposition.

Transcript

00:01 Okay, we've been given this fraction and we'd like to put it into partial fraction form.

00:04 So first off, we'll have to factorise the denominator here.

00:09 So you can see you can put an x cubed out of that.

00:11 So we're going to have one fraction of x, one fraction of x squared, one fraction of x cubed, and we're going to be left with here x plus 2.

00:17 So we'll need four fractions, a over x cubed, plus a over x squared, plus a over x squared, which that's supposed to be a b or whatever some sort of constant that we will work out soon c over x and then d over x plus two okay so expanding it out we know that 4x squared minus x minus two is going to be equal to a lot of x plus two plus b lots of x x plus 2 plus c lots of x squared x plus 2 and then plus d lots of just x cubed and we can see we can immediately work out a there by plugging in x as 0 you see that anything times by 0 is going be 0 so there'll be 0 here that's going to be 0 here it's going to be 0 0 0 so putting in 0 on this side we're only going to be left with minus 2 and plug in 0 here we're going to have 2a's.

01:32 So divided by 2.

01:33 A is clearly going to be equal to minus 1.

01:39 Equally, if we would put x in as minus 2, well, this bit would be 0a's, this bit would be bx times by 0, and this bit would be cx squared times by 0.

01:49 So we'll test in a minus 2.

01:51 Minus 2 cube is obviously minus 8d.

01:56 And here we'll have minus 2 squared...

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