Chapter 7.3, Problem 66E

To determine

To write: the form of the partial function decomposition of the rational expression and check result algebraically by combining fractions and check result graphically by using a graphing utility to graph the rational expression and the partial fraction in the same viewing window.

Answer to Problem 66E

  $\begin{array}{l}\frac{4x^4}{{(2x-1)}^3}=\frac{x}{2}+\frac{3}{4}+\frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}.\\ \end{array}$

Explanation of Solution

Given information: Given rational function is $\frac{4x^4}{{(2x-1)}^3}.$

Calculation:

Given function is,

  $\frac{4x^4}{{(2x-1)}^3}.$

Since the degree of the numerator is not less than degree of the denominator perform polynomial long division.

  $\frac{4x^4}{{(2x-1)}^3}=\frac{x}{2}+\frac{3}{4}+\frac{6x^2-4x+\frac{3}{4}}{{(2x-1)}^3}.$

Now, deal with $\frac{6x^2-4x+\frac{3}{4}}{{(2x-1)}^3}$ .

The form of the partial fraction decomposition is,

  $\begin{array}{l}\frac{6x^2-4x+\frac{3}{4}}{{(2x-1)}^3}=\frac{A}{(2x-1)}+\frac{B}{{(2x-1)}^2}+\frac{C}{{(2x-1)}^3}.\\ \frac{6x^2-4x+\frac{3}{4}}{{(2x-1)}^3}=\frac{{(2x-1)}^{2}A+(2x-1)B+C}{{(2x-1)}^3}.\\ $ 6x^2-4x+\frac{3}{4}={(2x-1)}^{2}A+(2x-1)B+C.$6x^2-4x+\frac{3}{4}=4x^{2}A+x(-4A+2B)+A-B+C.\end{array}$

Since both sides are in standard form, can equate the coefficients to come up with a system of equations.

  $\begin{array}{c}4A=\mathrm{6...........}(1)\\ -4A+2B=-\mathrm{4...........}(2)\\ A-B+C=\frac{3}{4}\mathrm{............}(3)\\ \text{Solving}\text{these}\text{equations}.\\ A=\frac{3}{2}.\\ B=1.\\ C=\frac{1}{4}.\end{array}$

Recombine to check algebraically.

  $\begin{array}{l}\frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}=\frac{6{(2x-1)}^2+4(2x-1)+1}{4{(2x-1)}^3}.\\ \frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}=\frac{24x^2-24x+6+8x-4+1}{4{(x-1)}^3}.\\ \frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}=\frac{24x^2-16x+3}{4{(x-1)}^3}.\\ \frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}=\frac{6x^2-4x+\frac{3}{4}}{{(x-1)}^3}.\\ \frac{4x^4}{{(2x-1)}^3}=\frac{x}{2}+\frac{3}{4}+\frac{3}{2(2x-1)}+\frac{1}{{(2x-1)}^2}+\frac{1}{4{(2x-1)}^3}.\\ \end{array}$

Graph of the original function and the partial fraction decomposition is given below.

  

Graph of the original function and the partial fraction decomposition. They’re both there and overlap because they’re equal.