Find the partial fraction decomposition of the rational function. (-3 x^2-3 x+27)/((x+2)(2 x^2+3

Find the partial fraction decomposition of the rational...

Key Concepts

Rational Functions

Rational functions are expressions defined by the ratio of two polynomials. They are important in many areas of mathematics because understanding their structure and behavior often leads to simplifications that facilitate integration, differentiation, and solving equations.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to rewrite a rational function as the sum of simpler fractions whose denominators are factors of the original denominator. This technique is especially useful when integrating rational functions or solving differential equations.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its irreducible factors, which could be linear or quadratic. This process is essential in partial fraction decomposition because it allows one to identify the correct form for the decomposed terms, ensuring that each fraction corresponds to a factor of the original denominator.

Decomposition Setup

Setting up the decomposition involves writing the rational function as a sum of fractions where each denominator is one of the factors identified through factorization. The numerators are assigned forms appropriate to the type of factor—constants for linear factors and linear expressions for irreducible quadratic factors.

Clearing Denominators and Coefficient Matching

After setting up the partial fraction form, the next step is to multiply through by the common denominator to eliminate fractions. This results in a polynomial equation where the coefficients of corresponding powers of the variable can be equated. Solving these equations yields the unknown constants in the decomposition.

Transcript

00:01 Okay, we've been giving this fraction and we want to find it in its final, put it into its partial fraction form.

00:08 So first off we need to take all the factors of the denominator and i think you can see this can be factorized so that we have x plus 2.

00:18 2x minus 3, x plus 3, minus 3 times 3 is minus 9, 3 times 2x.

00:26 We'll give a 6x minus 1 lot of x.

00:29 We're going to have a over x plus 2 plus b over 2x minus 3 plus c over x plus 3 okay and this means that we know since we'll have to times a by these b by these and c by these we know that that is going to be equal to this here so we have a times by 2x minus 3 times by x plus 3 plus b lots of x plus 2 x plus 3 plus c lots of x plus 2 x minus 3.

01:23 And we know this is always going to be equal to minus 3x squared, minus 3x plus 27, minus 3x plus 27.

01:37 Oakley dolly.

01:39 So how we're going to do with this is say if x is minus 3, well, i'm going to have 0 b's, because there'll be a 0 here, any times 0 0 ,000, and we also have 08.

01:46 And we don't be left with our cs, and we could work it out nice and easily.

01:51 So let's just do that.

01:52 Let's start with minus 3.

01:53 These are going to be 0.

01:55 This is going to be 0.

01:55 We're going to have minus 3 plus 2, which is going to give us minus 1.

02:03 So we'll have minus 1.

02:06 And minus 3 times by 2, minus 3 is going to give us minus 9.

02:11 So we're going to actually have a positive 9 lots of c.

02:14 We'll do that here, positive 9 lots of c.

02:17 And it's equal to minus 3 lots of minus 3 squared.

02:24 Which is basically minus three cubes so that's going to be equal to minus 27 minus three lots of minus three is good so what to do out here is plus nine and it's going to be plus 27 so plus nine plus 27 those will cancel we'll have nine c is equal to nine minus 37 plus 27 uh five five nine hopefully our c is equal to one doky -dokey.

02:57 Let's do x equal to minus two now so that we are only left with our a values...

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