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

Name: Problem 14, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T03:10:02-08:00
Duration: 5 min 41 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (2 x^2+x)/((x-1)^2(x+1)^2)

Find the partial fraction decomposition. (2...

Key Concepts

Repeated Factors

When a rational function's denominator has repeated factors, the decomposition must include a series of terms for each repeated factor, with each term corresponding to a higher power of the factor. This ensures that the decomposition accurately represents the original function.

Degree Comparison and Proper Fractions

A critical step in partial fraction decomposition is verifying that the rational function is proper—that is, the degree of the numerator is less than the degree of the denominator. If it is not, polynomial division must be performed prior to decomposition to ensure the function is expressed in a decomposable form.

Partial Fraction Decomposition

Partial fraction decomposition is the technique used to express a rational function—especially one whose denominator factors into simpler components—as a sum of simpler, easily integrable fractions. This is useful for integration and simplifying complex expressions.

Rational Functions

Rational functions are expressions that are the ratio of two polynomials. Understanding their structure, particularly the factorization of their denominator, is essential in applying decomposition techniques and analyzing their behavior.

Transcript

00:01 We have to find out the partial fraction decomposition of 2x squared plus x divided by x minus 1 whole square x plus 1 whole square, right? so the first thing we do is we write in a form of partial fractions.

00:23 So 2x square plus x is plus x minus 1 whole square plus 1 whole square is equal to a by x minus 1 plus b by x minus 1 whole square plus c by x plus 1 plus 1 plus d by x plus 1 whole square now what we can do is yes so now what you have to do is to cancel up the nominers we take this to the right side because this is in the partial fraction form we only have to find a, b, c, and d.

01:14 So we take this to the word side, and we get a, x minus 1, x plus 1, whole square, plus b, x plus 1 plus c, x minus 1, whole square, x plus d, which is going to be, well, this is going to be square here.

02:05 Here, d is going to be x minus 1 whole square.

02:12 So now what you have to do to find out, and all of this is equal to 2x squared plus x.

02:22 Now to find this out, to find the answer out, all you have to do is first things first.

02:31 You have to first of all find out two of these, right? to find, let's just say, to find d.

02:39 What we do is we put x is equal to minus 1.

02:44 When we put x equals a minus 1, this becomes 0, so this becomes 0, this becomes 0, this becomes 0, this becomes 0, this becomes 0, this becomes 0, so it'll only be left with d and a number.

02:56 So that way, you get the value of d from this.

03:00 So let me just say you want to find the value of b...

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