1-6 . Write out the form of the partial fraction...

$1-6$ . Write out the form of the partial fraction decomposition of
the function (as in Example $6 ) .$ Do not determine the numerical
values of the coefficients.
$$(a) \frac{x^{4}-2 x^{3}+x^{2}+2 x-1}{x^{2}-2 x+1} (b) \frac{x^{2}-1}{x^{3}+x^{2}+x}$$

Key Concepts

Repeated Linear Factors

When the denominator of a rational function contains repeated linear factors, the partial fractions are set up to include a separate term for each power of the factor. For example, if a factor is repeated k times, terms are included with denominators raised to each power from 1 to k.

Partial Fraction Decomposition

This is an algebraic technique used to express a complex rational function as a sum of simpler fractions. The idea is to break the function into parts that are easier to integrate, differentiate, or otherwise manipulate, with each part corresponding to a factor of the denominator.

Polynomial Long Division

When the degree of the numerator is greater than or equal to the degree of the denominator, polynomial long division is applied to rewrite the function as a polynomial plus a proper rational function. This simplifies the process of decomposition into partial fractions by ensuring that the remainder has a lower degree than the denominator.

Irreducible Quadratic Factors

An irreducible quadratic factor is a quadratic that cannot be factored further over the reals. In the context of partial fraction decomposition, each irreducible quadratic factor is associated with a numerator that is a linear expression, ensuring that the degree of the numerator is one less than the quadratic denominator.

Transcript

00:01 For part a, we need to find the decomposition of this expression.

00:17 Now, the first thing we always do before we find a decomposition is to check if this is an improper fraction or not, and it is not because the degree of the top, 4 is greater than the degree of the bottom, too.

00:36 So we need to rewrite this as an improper fraction.

00:41 Now, there are multiple ways to do that.

00:45 One that always will work is doing long division until you get one function plus an improper fraction.

00:55 However, sometimes you may want to look out for an easier way.

01:00 In this case, we want to notice is this part here, x the fourth to x squared here, that part, is a multiple.

01:11 Is a multiple essentially of this denominator.

01:16 So we could rewrite it so that we could split this function up, as i will show you, and then cancel out certain parts.

01:22 So basically, this would be equal to x -to -the -fourth minus 2 x -cube plus x squared.

01:32 So i'm splitting it up at that part, and then we have the rest of the numerator.

01:38 So we have plus 2x minus 1 over that same denominator.

01:53 Basically what i was saying before is we can factor out from this numerator x squared, and you'll see what happens.

02:07 In these brackets, we have what's in the denominator.

02:12 Therefore, those can just cancel out like this, leaving us with x squared.

02:21 And then we have this other part, 2x minus 1, and and also notice this is an improper fraction.

02:32 The degree of the top is one, the degree of the bottom is two, so it can be decomposed as we will do...

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