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

Name: Problem 26, Chapter 8: College Algebra
Uploaded: 2020-06-20T21:56:36-08:00
Duration: 4 min 58 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (x^2+x)/((x+2)(x-1)^2)

Find the partial fraction decomposition of each rational...

Key Concepts

Repeated Factors

When a factor in the denominator appears multiple times, each occurrence up to its multiplicity must be accounted for in the decomposition. This means including separate terms for each power of the repeated factor, ensuring that the overall decomposition accurately reflects the structure of the original rational expression.

Solving Systems of Equations

After setting up the decomposition with unknown coefficients for each term, the next step is to solve for these coefficients. This is typically done by clearing the denominators and equating the coefficients on both sides of the resulting polynomial identity, which leads to a system of linear equations that can be solved using algebraic methods.

Partial Fraction Decomposition

Partial fraction decomposition is an algebraic method that expresses a complex rational function as a sum of simpler fractions. This approach is particularly useful for integration and solving equations, as each of the simpler fractions is easier to work with than the original complex fraction.

Factorization of the Denominator

Successful partial fraction decomposition starts with fully factoring the denominator. Recognizing linear and irreducible quadratic factors, as well as their multiplicities, ensures that the decomposition form is set up correctly with distinct or repeated terms corresponding to each factor.

Transcript

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

00:07 Now, in order to split this up, we could say that sum number a over x plus 2, that's going to be our non -repeated function plus.

00:19 Now, our repeated function is x minus 2 squared, so that's going to be some number b over x minus 1 plus some number c over x minus 1 square.

00:29 Now let's multiply everything out by the common denominator of x plus 2 times x minus 1 squared.

00:36 To get x squared plus x is equal to a times x minus 1 squared plus b times x plus 2 times x minus 1 plus c times x plus 2.

01:03 Now after multiplying all of that out and grouping like terms, we will get the identity.

01:11 X squared plus x is equal to a plus b x squared plus negative 2a plus b plus c times x plus a minus 2b plus 2c plus 2c.

01:42 Now let's equate the coefficients on both sides of the equal sign.

01:46 We'll get a plus b equals 1.

01:51 Because there's a, we can say there's a 1 here.

01:55 This will equal 1.

01:57 Save for x.

01:58 There's a 1 in front of the x.

01:59 So we could say that negative 2a plus b plus c equals 1.

02:07 Lastly, a minus 2b plus 2c will equal 0.

02:12 There's no other terms on the left side.

02:15 So now we have three equations and three unknown so we can solve for a, b, and c.

02:19 Now, let's look, both two equations equal one...

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