`(16x^4)/(2x - 1)^3` Write the partial fraction decomposition of the improper rational expression.

Expert Answers

An illustration of the letter 'A' in a speech bubbles


Since the rational expression is an improper expression, we have to express the fraction as a sum of simpler fractions with the degree of the polynomial in the numerator less than the degree of the polynomial in the denominator.

Let's expand the denominator of the rational expression,


Dividing using the long division method yields,


Since the polynomials do not completely divide, so we have to continue with the partial fractions of the remainder expression,

Let, `(24x^2-16x+3)/(8x^3-12x^2+6x-1)=A/(2x-1)+B/(2x-1)^2+C/(2x-1)^3`





equating the coefficients of the like terms,

`4A=24`                     ----- equation 1

`-4A+2B=-16`    ------ equation 2

`A-B+C=3`           ------ equation 3

From equation 1,



Plug the value of A in equation 2,





Plug the values of A and B in equation 3,







Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial