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

Name: Problem 16, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T04:10:54-08:00
Duration: 4 min 22 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (4 x^3+3 x^2+5 x-2)/(x^3(x+2))

Find the partial fraction decomposition. (4 x^3+3 x^2+5...

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational function as a sum of simpler fractions whose denominators are factors of the original denominator, which makes the function easier to integrate or manipulate. This method breaks down a complex fraction into a series of simpler terms that are easier to work with in further mathematical operations.

Proper Rational Functions

A proper rational function is one where the degree of the numerator is less than the degree of the denominator. If the function is improper, polynomial division may first be required. In this context, after ensuring that the function is proper, partial fractions can be applied reliably.

Factorization of the Denominator

Accurately factoring the denominator is crucial since the factors determine the form of the partial fractions. The denominator must be decomposed into irreducible factors over the reals, including linear factors and possibly irreducible quadratic factors, which then guide the setup of the decomposition.

Repeated Linear Factors

When the denominator contains repeated factors, such as a term raised to a power greater than one, each occurrence must be accommodated in the partial fraction setup. For a repeated linear factor like x^n, individual terms are written for each power, from 1 to n (for example, A/x, B/x^2, ...).

Matching Coefficients and System of Equations

Once the partial fractions are set up with unknown coefficients, both sides of the resulting equation are multiplied by the common denominator and then expanded to form a polynomial identity. The coefficients of corresponding powers of the variable are then equated, resulting in a system of equations that can be solved to determine the unknown constants.

Transcript

00:01 So here we have to find the partial fraction decomposition of 4x cube plus 3x square plus 5x minus 2 divided by x cubed x plus 2.

00:25 Now first things first right square here is we're going to divide this into its factors and we're going to write into partial fraction form that is 4x cube plus 3x square plus y -ex minus 2 divided by x cube x plus 2 is equal to a by x plus b by x square plus c by x cube plus d by x plus 2 and these are the partial fractions we have to find the value of a b c and d and we'll take this denominator to the right side to find the values when we do this that what we get is 4x cube plus 3x square plus 5x minus 2 is equal to a x square x plus 2 plus b x plus 2 plus b x plus 2 plus c x plus 2 plus d x now to find the values are a, b, and c, a, b, c, and d, the constant, what do you do is you plug in the roots of these factors.

02:00 So one root is zero.

02:02 X is equal to zero.

02:04 So when we do that, we get the value of c, right? we get the value of c, which would be minus one.

02:26 Yeah, that would be minus one.

02:29 Right.

02:30 Then we plug in the value of the other root, which is minus two.

02:34 You get the value of d, which is 4.

02:46 Well, x is equal to minus 2, d is equal to 4.

02:51 So you get these values.

02:55 Now, if we get these values, what do you do is you plug them in this equation, and since you cannot find any other roots to find the constants, what do you do is you expand them, and you collect their powers...

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