Find the partial fraction decomposition of (4 x^2+5 x-9)/(x^3-6 x-9)

Name: Problem 65, Chapter 7: Precalculus
Uploaded: 2020-04-15T00:18:51-08:00
Duration: 9 min 16 s
Channel: Ashton Collins
Description: Find the partial fraction decomposition of (4 x^2+5 x-9)/(x^3-6 x-9)

Find the partial fraction decomposition of (4 x^2+5...

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function, which is the ratio of two polynomials, into a sum of simpler fractions. This technique facilitates further analysis such as integration, Laplace transforms, or solving differential equations by decomposing difficult expressions into manageable parts.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of its irreducible factors. This step is essential in partial fraction decomposition because identifying the factors in the denominator determines the structure of the simpler rational expressions that sum to the original rational function.

Rational Functions

Rational functions are functions defined by the ratio of two polynomials. They are a fundamental concept in algebra and calculus and require special techniques for analysis and computation, such as partial fraction decomposition, especially when dealing with integration or solving equations.

Transcript

00:01 So we're asked to find the partial fraction decomposition of the expression.

00:05 We're giving 4x squared plus 5x minus 9 over x cubed minus 6x minus 9.

00:11 Now, we need to factor the bottom.

00:14 Now, because it's a trinomial and it's got a degree of 3, the only really way we can factor this is we're going to use possible rational root theorem.

00:27 So you look at x cubed minus 6x, minus 6x, minus.

00:37 Nine, we've got to find all the possible roots.

00:41 So we'll find all the possible factors of a and c.

00:49 So factors of this are plus or minus one.

00:54 Factors of nine are plus or minus one, plus or minus three, and plus or minus nine.

01:06 And then all the possible factors, our roots, are going to be these over the, these.

01:17 So what we get are plus or minus one, plus or minus three, and plus or minus nine are all the possible routes.

01:28 So the way we find these is if we plug any of these values in, if we get zero, that that will give us the root.

01:38 So what happens in is i plug in three.

01:46 It'll show that that's a root, which means there's a factor of x minus x minus 3.

01:59 So now i'm going to take this x minus 3 and divide it into x cubed minus 6x minus 9.

02:08 And what i'm going to use, i'm going to use synthetic division.

02:10 So i'll use my k.

02:12 So if it's x minus 3, the k is going to be 3...

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