Sure, I'll guide you through the process of finding the partial fraction decomposition for each of these expressions:
1. For the fraction [tex]\(\frac{5x-3}{(x^2 - x + 3)(x^2 - 3x - 1)}\)[/tex]:
[tex]\[
\frac{5x-3}{(x^2 - x + 3)(x^2 - 3x - 1)} = \frac{13x - 20}{18(x^2 - 3x - 1)} - \frac{13x + 6}{18(x^2 - x + 3)}
\][/tex]
2. For the fraction [tex]\(\frac{x^2 + 2x - 5}{(2 - 5x - x^2)(x^2 - x - 1)}\)[/tex]:
[tex]\[
\frac{x^2 + 2x - 5}{(2 - 5x - x^2)(x^2 - x - 1)} = -\frac{3(7x + 43)}{41(x^2 + 5x - 2)} + \frac{21x - 38}{41(x^2 - x - 1)}
\][/tex]
3. For the fraction [tex]\(\frac{x-2}{x^2 + 3x + 2}\)[/tex]:
First, factor the denominator: [tex]\(x^2 + 3x + 2 = (x + 1)(x + 2)\)[/tex].
[tex]\[
\frac{x-2}{(x + 1)(x + 2)} = \frac{4}{x + 2} - \frac{3}{x + 1}
\][/tex]
4. For the fraction [tex]\(\frac{x^3 - x^2 - 3x}{(x^2 + 6x - 2)(2 - x^2)(2x^2 - 7x + 1)}\)[/tex]:
[tex]\[
\frac{x^3 - x^2 - 3x}{(x^2 + 6x - 2)(2 - x^2)(2x^2 - 7x + 1)} = -\frac{12x + 19}{438(x^2 - 2)} + \frac{54x + 385}{762(x^2 + 6x - 2)} - \frac{806x - 2141}{9271(2x^2 - 7x + 1)}
\][/tex]
5. For the fraction [tex]\(\frac{x^2 + 2x - 5}{(4 - x^2)(2x^2 - 3x + 1)}\)[/tex]:
Notice that [tex]\(4 - x^2\)[/tex] can be rewritten as [tex]\((2 - x)(2 + x)\)[/tex].
[tex]\[
\frac{x^2 + 2x - 5}{(4 - x^2)(2x^2 - 3x + 1)} = \frac{2}{2x - 1} - \frac{1}{12(x + 2)} - \frac{2}{3(x - 1)} - \frac{1}{4(x - 2)}
\][/tex]
These are the partial fraction identities for each of the fractions provided.
