Find the partial fraction decomposition. (9 x^2-3 x+8)/(x^3+2 x)

Name: Problem 19, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T04:29:58-08:00
Duration: 2 min 42 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (9 x^2-3 x+8)/(x^3+2 x)

Find the partial fraction decomposition. (9 x^2-3...

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler rational expressions. This method is particularly useful in integration, solving differential equations, and simplifying complex rational expressions by breaking them down into components with simpler denominators.

Factorization of Polynomials

Factorization of polynomials involves expressing a polynomial as a product of its irreducible factors. In the context of partial fractions, correctly factoring the denominator is essential because it determines the form of the partial fractions and the subsequent method for finding the constants in the decomposition.

Rational Functions

Rational functions are ratios of two polynomials. Understanding the degrees of the numerator and denominator is important because it influences whether polynomial long division is needed and how the partial fraction decomposition will be set up.

Equating Coefficients

Equating coefficients is a method used to determine unknown constants in an equation by matching the coefficients of corresponding powers of variables on both sides of an equation. This technique is crucial in the process of partial fraction decomposition to systematically solve for the unknowns in each term.

Transcript

00:01 So we're redoing this video because there's a typing unit in the last one.

00:05 So let's just look at the partial fraction of decomposition that we have to do in this.

00:09 So we have 9x square minus 3x plus 8 divided by x cube plus 2x.

00:14 So the first thing we do is we take x and x common in the denominator.

00:19 So you get x into x squared plus 2.

00:22 That's straightforward.

00:23 We're not looking at the numerator right here.

00:24 Just don't look down.

00:26 Right.

00:26 So now what we do is we can rewrite this, rewrite this.

00:31 As this equal to the partial fraction a x plus b by x code list 2 plus c by x this is how you write partial fractions right this is this whole fraction is a sum of these two fractions so now to find this find a b and c what we do is we remove the denominators from both sides so we take this to the right side just we cancel out the denominators and you get something like 9x minus 3x plus 8 is equal to a x square plus bx plus cx 4 plus 2 so to find the values of their constants what you have to do is you have to plug in the values of x the roots so there's x x square and x squared 2 so there means that that means that there's only one root which is x equal to 0 so these two cancel out you only get c and c when x equals 0 is 4 by common logic.

01:34 And then to find the values of a and b, you cannot do that where roots.

01:37 So what do you do is you collect the powers.

01:43 So let's just look at 9x square, right? so that means that everything that sums up to x squared, the coefficients of that have to be equal to 9...

Please give Ace some feedback