Find the partial fraction decomposition. (2 x^4-2 x^3+6 x^2-5 x+1)/(x^3-x^2+x-1)

Name: Problem 23, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T20:36:09-08:00
Duration: 3 min 37 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (2 x^4-2 x^3+6 x^2-5 x+1)/(x^3-x^2+x-1)

Find the partial fraction decomposition. (2 x^4-2 x^3+6...

Key Concepts

Improper Rational Functions

An improper rational function is one in which the degree of the numerator is equal to or greater than the degree of the denominator. In such cases, it is necessary to perform polynomial long division to rewrite the expression as a polynomial plus a proper rational function before applying other techniques.

Polynomial Long Division

Polynomial long division is used when dividing polynomials, especially when the degree of the numerator is not less than the degree of the denominator. This process separates the rational function into a sum of a polynomial and a proper fraction, which simplifies further analysis and decomposition.

Factorization of Polynomials

Factorization involves expressing a polynomial as a product of its irreducible factors. In the context of partial fraction decomposition, the denominator must be factored into linear and/or irreducible quadratic factors because these factors determine the structure and form of the partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition is the process of breaking down a complex rational expression into a sum of simpler fractions, each with a simpler denominator. This method simplifies the analysis and integration of rational functions by isolating terms associated with the individual factors of the denominator.

Transcript

00:02 Hello, so today i had to leave this video partway because of some issues, but let's just continue i'm going to start from the beginning.

00:10 We have to find the partial fraction decomposition of this whole fraction 2x raised 4 minus 2x cube plus 6x squared minus 5x plus 1 divided by x cubed minus x squared plus x minus 1.

00:21 So you can see that the degree of the numerator is greater than degree of a denominator right it should be less than it should be well, one less than degree of denominator.

00:34 So what we will do here is we'll divide it.

00:38 So we do long division of this with this, and we get this fraction here.

00:44 2x plus 4x square minus 3x plus 1 divided by x cubed minus x squared plus 1.

00:50 So now what we do is we divide this into factors, 4x2 minus 3x plus 1, divided by x squared plus 1, x minus 1.

01:00 And we write them in the partial fraction form.

01:02 A x plus b divided by x squared plus one plus c and divided by x minus one so what you'll do is to find a b and c to find the partial fraction you take this to the right side you take the denominator to the right side once you do that you cancel these denominators out and you get something like four x squared minus three x plus one is equal to a x x minus one plus b x minus one plus c x squared plus one now to find the values of ab and c we'll use the roots of x and x and these are equations.

01:34 So the roots will be first of all x equals 1 is 1 root.

01:37 It will make this term and this term 0.

01:41 So xx squaredly will be the only term left.

01:46 So when we do that we get answers equal to 2c is equal to 2.

01:50 That means c is equal to 1...

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