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

Name: Problem 39, Chapter 10: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-21T18:52:50-08:00
Duration: 5 min 20 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (x^2)/(x^3-4 x^2+5 x-2)

Find the partial fraction decomposition of each rational...

Key Concepts

Partial Fraction Decomposition

This is a method used in algebra and calculus to express a complicated rational function as a sum of simpler fractions, which are easier to integrate or manipulate. It involves breaking down the original rational expression by factoring its denominator and then expressing the result as a sum involving constants or simpler polynomial expressions.

Polynomial Factorization

This is the process of breaking down a polynomial into the product of its irreducible factors. In partial fraction decomposition, factoring the denominator is crucial because it allows you to determine the form of each term in the decomposition. The linear or quadratic factors obtained dictate the structure of the terms in the final expression.

Degree Comparison in Rational Functions

Before applying partial fraction decomposition, it is important to compare the degree of the numerator with the degree of the denominator. If the numerator has a degree equal to or greater than the denominator, polynomial long division must first be applied to rewrite the expression so that the numerator's degree is less than the denominator's degree.

Equating Coefficients

This technique is used to determine the unknown constants in the partial fraction decomposition. Once the rational expression is rewritten as a sum of simpler fractions, the numerators are manipulated and the coefficients for each power of the variable are equated across both sides of the equation to solve for these unknowns.

Transcript

00:01 Let's find the partial fraction decomposition of this rational expression by first looking at the denominator.

00:07 Now, the denominator of this expression can be as factorable.

00:11 So we can rewrite this as x squared over x minus 1 squared times x minus 2.

00:19 So we have one repeating and one non -repeating function in our denominator.

00:23 So our partial fraction decomposition will be sum number a over x minus 1 plus some number b over x minus 1 squared plus some number c over x minus 2.

00:39 Now let's multiply both sides by the denominator x minus 1 squared times x minus 2.

00:46 The result will be x squared is equal to a times x minus 1 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 minus x minus 1 square.

01:04 Now, expand and multiply everything out on the right side of the equation and group -like terms.

01:10 And when doing so, you will come up with the identity of x squared being equal to a plus c times x squared plus negative 3a plus b minus 2c times x times x, plus 2a minus 2b plus c.

01:39 So now that we have grouped all of our terms, let's equate the coefficients on both sides of the equal side.

01:48 So let's see, we have 1x squared, therefore a plus c is going to equal 1.

01:56 There's no other terms on the left, therefore negative 3a plus b minus 2c is equal.

02:04 To 0 and 2a minus 2b plus c is equal to 0.

02:12 Now let's solve for b first by using substitution.

02:17 Let's let a equal 1 minus c...

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