Write the partial fraction decomposition of the given rational expression. startfraction 5 x plus 2 over x superscript 4 baseline minus 4 x squared endfraction

Answer:Step-by-step explanation:Given is a rational algebraic expression in x as[tex]\frac{5x+2}{x^4-4x^2}[/tex]First let us factorize the denominator.[tex]x^2(x^2-4)\\=x^2(x+2)(x-2)[/tex]Let the given term be[tex]\frac{A}{x} +\frac{B}{x^2} +\frac{C}{x-2} +\frac{c}{x+2} \\\\=\frac{Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)}{x^4-4x^2}[/tex]Equate the numerator to get[tex]Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)[/tex]=[tex]5x+2[/tex]Substitute x =-2[tex]4D (-4) = -8\\D=\frac{1}{2}[/tex]Now substitute x =2We get[tex]16C=12\\C=\frac{3}{4}[/tex]Put x=0[tex]-4B=2\\B=\frac{-1}{2}[/tex]Next equate X cube terms to 0 on right side[tex]A+C+D=0\\A=\frac{-5}{4}[/tex]Hence partial fraction is[tex]\frac{-1}{2x^2} -\frac{5}{4x} +\frac{1}{2(x+2)} +\frac{3}{4(x-2)}[/tex]