Find the partial fraction decomposition of each rational expression. (1)/(x^3-8)

Name: Problem 21, Chapter 8: College Algebra
Uploaded: 2020-06-20T20:25:58-08:00
Duration: 6 min 1 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (1)/(x^3-8)

Find the partial fraction decomposition of each rational...

Key Concepts

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of its irreducible factors. In the context of partial fractions, this step is critical because the denominator must be factored completely, often using special product formulas such as the difference or sum of cubes, to determine the appropriate form of the partial fraction decomposition.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler rational expressions whose denominators are factors of the original denominator. This technique is particularly useful for integration and solving differential equations, as it transforms a complex rational expression into a collection of easier-to-manage pieces.

Irreducible Quadratic and Linear Factors

When a polynomial is factored, its factors can be either linear or irreducible quadratic factors, and each type is handled differently in the decomposition process. For a linear factor, the corresponding term in the partial fraction expansion typically has a constant numerator, while for an irreducible quadratic factor, the numerator is generally a linear polynomial. Recognizing and handling these types correctly ensures the decomposition accurately reflects the original rational expression.

Transcript

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

00:06 Now, x cubed minus 8 is factorable.

00:09 We could factor an x minus 2 out.

00:13 When we do that, we're left with x squared plus 2x plus 4.

00:21 All of this under 1.

00:25 Now, for our partial fraction decomposition, let's look at what we have in the denominator.

00:29 Now, x minus 2 is a non -repeating function.

00:33 So that will be some non -repeating function.

00:35 So that will be some number a over x minus 2 plus.

00:40 Now our section function is a non -repeating irreducible function.

00:46 So that will be written as sum number b times x plus some number c all over our non -repeating irreducible function, x squared plus 2x plus 4.

01:00 Now let's multiply both sides of the equal sign here with the denominator x minus 2 times x squared plus 2x plus 1 to yield 1 equals a times x squared plus 2x plus 4 plus bx plus c times x minus 2.

01:28 Now let's multiply everything out on the right side so we'll get 1 equals a x squared plus 2 a x squared plus 2 a x squared.

01:38 Plus 4a plus bx squared minus 2b x plus cx minus 2c.

02:01 Now let's group like terms.

02:06 So we have a x squared bx squared so that's going to be a plus b times x squared plus what's in front of x's is.

02:16 We have one there, one there, and one there.

02:19 So that's plus 2a minus 2b plus c times x.

02:28 And then we have a plus 4a and a minus 2c.

02:31 So plus 4a minus 2c.

02:37 Now let's equate the coefficients on both sides of the equal side...