Find the partial fraction decomposition of each rational expression. (2 x+3)/(x^4-9 x^2)

Name: Problem 45, Chapter 8: College Algebra
Uploaded: 2020-06-21T18:22:12-08:00
Duration: 7 min 45 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (2 x+3)/(x^4-9 x^2)

Find the partial fraction decomposition of each rational...

Key Concepts

Partial Fraction Decomposition

A technique in algebra used to break a rational function into a sum of simpler rational expressions with denominators that are typically linear or irreducible quadratic factors. This method is especially useful in integration and solving differential equations because it simplifies the function into manageable components.

Polynomial Factorization

The process of expressing a polynomial as a product of its factors. In the context of partial fractions, factorization is necessary to decompose the rational expression since the factors of the denominator determine the form and number of simpler components in the decomposition.

Rational Expressions

These are fractions where both the numerator and denominator are polynomials. Understanding their structure and properties, including how to simplify and factor them, is essential for applying partial fraction decomposition effectively.

Transcript

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

00:07 Now here we can see that x squared is common between x -4th minus 9x squared.

00:12 So we could factor an x squared out of x -the -fourth minus 9x squared.

00:20 So if we take an x -squared out, we're left with x -squared minus 9.

00:25 Now, x -squared minus 9 can still be factored.

00:30 X squared minus x minus x minus 3 times x plus 3.

00:33 So we could rewrite this as 2x plus 3 over x squared times x minus 3, x plus 3.

00:43 Now we are ready to start writing our partial fraction decomposition.

00:47 So if you notice x squared is repeating, so we're going to write sum number a over x plus some number b over x squared.

00:59 Plus now we have our two non -repeating functions plus some number c over x minus 3 plus some number d over x plus 3 okay now we're ready to try to solve for a b c and d let's multiply both sides of the equal sign by the common denominator x squared times x squared minus 9 and the result will be 2x plus 3 is equal to a x so we'll multiply x squared minus 9 is x minus 3 times x plus 3 plus b times x minus 3 x plus 3 plus c x squared x plus 3 plus d x squared x minus 3 now let's expand, expand everything on the right, multiply everything out, and then group our like terms.

02:16 And when you do that, you're left with the identity.

02:19 2x plus 3 is equal to a plus c plus d x cubed plus b plus 3c plus...

02:44 Minus plus 3c minus 3d x squared plus negative 9 a x plus negative 9b.

03:08 Okay.

03:09 Now that we have grouped all our terms, let's equate the coefficients on both sides of the equal side.

03:14 Well, there's no x cubed on the left so we could write a plus b.

03:19 A plus c plus d is equal to zero.

03:27 There's no x squared, so we can write b plus 3c minus 3d is equal to zero.

03:37 There is another two x's on the left.

03:42 So we could say, we could write that negative 9a will equal 2.

03:47 Likewise, there's a 3 here.

03:49 Negative 9b, that's what will correspond to that.