Respuesta :
Answer:
[tex]\displaystyle\frac{A}{x} +\frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx+E}{x^2 + 5}[/tex]
Step-by-step explanation:
We are given the following fraction in the question:
[tex]\dfrac{x^4+6}{x^5 + 5x^3}[/tex]
W can decompose the given fraction in the following manner:
[tex]\dfrac{x^4 + 6}{x^5 + 5x^3}\\\\=\displaystyle\frac{x^4 + 6}{x^3(x^2 + 5)}\\\\\text{The decomposition takes place as}\\\\=\frac{A}{x} +\frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx+E}{x^2 + 5}\\\\=\frac{Ax^2+Bx+C}{x^3} + \frac{Dx+E}{x^2 + 5}\\\\=\frac{(x^2+5)(Ax^2+Bx+C) + x^3(Dx+E)}{x^3(x^2+5)}\\\\=\frac{x^3(Dx+E) + x^2(x^2+5)A + x(x^2+5)B + (x^2+5)C}{x^3(x^2+5)}[/tex]
