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

Name: Problem 23, Chapter 8: College Algebra
Uploaded: 2020-06-20T21:08:44-08:00
Duration: 7 min 46 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (x^2)/((x-1)^2(x+1)^2)

Find the partial fraction decomposition of each rational...

Key Concepts

Partial Fraction Decomposition

This is a method used to express a complex rational function as a sum of simpler fractions. It is particularly useful for integration and simplification of expressions by breaking down a fraction into more manageable parts with simpler denominators.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. Understanding these functions, particularly how the degrees of the numerator and denominator influence the decomposition process, is essential when applying partial fraction techniques.

Repeated Factors

When the denominator contains repeated factors, each occurrence must be accounted for in the decomposition. This means writing separate terms for each power of the repeated factor to ensure the complete representation of the original function.

Polynomial Factorization

Factorizing the denominator into its irreducible factors is a foundational step in the partial fractions process. It reveals the structure of the decomposition and helps determine the necessary form and number of terms in the partial fractions.

Solving Systems of Equations

After setting up the proposed partial fractions, equating coefficients or plugging in specific values leads to a system of linear equations. Solving these equations is crucial to finding the values of the unknown constants in the decomposition.

Transcript

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

00:06 Now, our denominator here has two repeating functions.

00:13 So, x minus 1 squared, we could rewrite as a over x minus 1 plus b over x minus 1 square.

00:24 Plus, now look at x plus 1 squared.

00:26 That'll be sum number c over x plus 1 plus some number d, over x plus 1 squared.

00:35 Now let's get rid of our fractions by multiplying everything by the denominator x minus 1 squared times x plus 1 squared.

00:42 So when you multiply everything by that, we're left with x squared is equal to a times x minus 1 times x plus 1 squared, plus b times x plus 1 squared, plus c times x minus 1 squared, squared x plus 1 plus d x minus 1 squared.

01:16 So multiply all of that out and group your like terms.

01:20 So if you multiply everything and then look for your x cubed, x squared, x squared, factor group terms, and you'll get x squared equals a plus c x cubed plus a plus b.

01:39 B minus c plus d x squared, times x squared, plus negative a plus 2b minus c minus 2d x plus what's left over, negative a plus b plus c plus d.

02:13 Now that we have group like terms, let's equate coefficients in order to get equations to solve for our variables, or a, b, c, and d.

02:25 So there's no x cubed on the left side so we could write a plus c is equal to zero.

02:32 There's one x square.

02:34 There's a coefficient one, right? we can say there's one here in front of x squared.

02:39 So a plus b minus c plus d is going to equal 1.

02:42 A plus b minus c plus d is equal to 1.

02:51 There's nothing.

02:52 There's no x on the left side.

02:53 There's no x term.

02:54 So we could just write negative a plus 2b minus c minus 2d is equal to 0.

03:03 And likewise, there's no other terms.

03:06 There's no other numbers.

03:07 So we could let our final expression of negative a plus b plus c plus d is equal to zero.

03:20 Okay, so now we have four equations and four unknowns to solve for.

03:24 Let's look at a plus c equals zero.

03:28 That implies that a equals negative c.

03:33 Okay, that's just to start with...

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