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

Name: Problem 20, Chapter 8: College Algebra
Uploaded: 2020-06-20T19:56:24-08:00
Duration: 5 min 16 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (x+1)/(x^2(x-2))

Find the partial fraction decomposition of each rational...

Key Concepts

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding their structure is important because it allows us to analyze properties such as domain restrictions, discontinuities, and simplification techniques, which are foundational for studying more advanced algebraic concepts.

Factoring Polynomials

Factoring polynomials is the process of writing a polynomial as a product of its irreducible factors. This step is crucial in partial fraction decomposition since the factors of the denominator determine the form of the simpler fractional components into which the original rational expression is decomposed.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator. This technique is particularly useful in integration, differential equations, and when performing inverse Laplace transforms, as it simplifies the expression into more manageable parts.

Handling Repeated Factors

When the denominator contains repeated factors, each repeated factor must be accounted for in the decomposition. The method requires setting up a series of terms where the repeated factor appears raised to different powers, ensuring that the decomposition is complete and that the original expression can be reconstructed from these components.

Clearing Fractions and Equating Coefficients

After expressing a rational expression as a sum of partial fractions, multiplying both sides of the equation by the common denominator removes all fractional terms. This step results in a polynomial identity where one can equate the coefficients of like powers of the variable on both sides to solve for the unknown constants involved in the decomposition.

Transcript

00:01 In order to start to find the partial fraction decomposition of our rational expression, let's look at the denominator.

00:07 We have a repeating and a non -repeating function.

00:12 So let's look at our repeating function.

00:14 So remember, if we have a function that repeats, our partial fraction decomposition is going to be some number a over x plus some number b over x squared.

00:27 Now we're going to add that to our non -repeating function.

00:32 That will be sum number c over x minus 2.

00:37 Now let's multiply both sides by the denominator of a rational express.

00:42 Just going to multiply both sides by x squared times x minus 2 for the result of x plus 1 equal to a times x minus 2 plus b times x minus 2 plus c times x minus 2 plus c times x minus 2 plus c times x squared.

01:17 Now let's multiply everything out to get, so we're going to get x plus 1 is equal to.

01:24 So a x times x minus 2 is a x squared minus 2 a x plus now b times x minus 2 is b x minus 2 b and plus c times x squared now let's group our terms on the right side.

01:50 Let's group our like terms on the right side of the equal sign.

01:53 So x plus 1 equals.

01:56 So let's distribute out in x squared.

01:59 So we have a x squared and cx squared.

02:02 So that's equals a plus c x squared.

02:08 Now let's look at our x's plus negative 2a plus b.

02:18 X and what's left minus 2b.

02:26 So now let's look at the coefficients on both sides of the equal side.

02:29 When we equate them, we're going to get, so what's in front of x? we have, what's the coefficient in front of x? you can say that could be a 1.

02:37 1x.

02:42 There we go.

02:43 So negative 2a plus b is going to equal 1...