Find the partial fraction decomposition of each rational expression. (x+1)/(x^2(x-2)^2)

Name: Problem 24, Chapter 8: College Algebra
Uploaded: 2020-06-20T21:18:09-08:00
Duration: 6 min 2 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (x+1)/(x^2(x-2)^2)

Find the partial fraction decomposition of each rational...

Key Concepts

Partial Fraction Decomposition

This concept involves expressing a complicated rational expression as a sum of simpler fractions whose denominators are factors of the original denominator. It is used to simplify calculations in integration, solving differential equations, and other algebraic manipulations.

Rational Expressions

Rational expressions are fractions where both the numerator and denominator are polynomials. The technique of partial fraction decomposition is applicable when the degree of the numerator is strictly less than the degree of the denominator, making the expression 'proper'.

Repeated Linear Factors

When the denominator of a rational expression includes repeated linear factors, each repeated factor requires a series of terms in the partial fraction decomposition, each with increasing powers of the factor. This ensures that the decomposition accurately reflects the structure of the original expression.

Method of Equating Coefficients

After setting up a partial fraction decomposition, corresponding coefficients of like powers on both sides of the equation are equated. This technique leads to a system of equations that can be solved to determine the unknown constants in the decomposition.

Transcript

00:01 In order to solve for our partial fraction decomposition of the rational expression, we must first look at the denominator.

00:06 Our denominator here has two repeating functions.

00:10 So we could rewrite this as sum number a.

00:13 This is our first repeating function plus some number b over x squared.

00:19 Now let's look at our second repeating function, x minus two squared.

00:22 So we plus some number c, x minus two, plus some number d over x minus two squared.

00:30 Now let's clear the denominator.

00:33 Denominators here.

00:34 Let's get rid of all these fractions by multiplying both sides by the denominator of our rational expressions.

00:39 We're ultimately both sides by x squared times x minus two square.

00:43 When we do that, we're left with x plus 1 is equal to a x times x minus 2 squared plus b times x minus 2 squared plus c x squared plus c x squared.

01:00 Times x minus 2 plus d x squared.

01:11 Now, if you multiply everything out on the right side and group your like terms, if you do that, you're going to get x plus 1 is equal to a plus c x cubed plus negative 4a plus b minus 2c.

01:41 Plus d x squared plus 4a minus 4b times x plus 4b plus 4b that's what's left what's left over is 4b so now when we equate the coefficients on both side of the equal sign we'll get the following so there's no x cubed on the left side so we could write a plus c is equal to zero there's no x squared on the left side either so we could rewrite the coefficient of front of x score on the right.

02:29 So we could write negative 4a plus b minus 2c plus d is equal to 0.

02:40 Now there is 1x.

02:42 The coefficient for an x on the left is 1.

02:43 So we could rewrite 4a minus 4b is equal to 1 and 4b is equal to 1.

02:55 We have 1 here and what's left is the 4b.

02:57 So 4b equals 1...

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