Partial Fraction Decomposition

11.5 PFD Case 3 & 4

Denominators with Quadratic Irreducible factors

Multiple Choice

Write the partial fraction decomposition initial set up for the following

$$\frac{3x^2+2}{x\left(x^2+3\right)}$$  

• $$\frac{A}{x}+\frac{Bx+C}{x^2+3}$$  

• $$\frac{A}{x^2}+\frac{Bx+C}{x^2+3}$$  

• $$\frac{A}{x}+\frac{Bx+C}{x^3+3x}$$  

Multiple Choice

Determine the partial fraction decomposition of

$$\frac{x^2+3}{x^3+1}$$  

• $$\frac{\frac{4}{3}}{x+1}+\frac{-\frac{1}{3}x+\frac{5}{3}}{x^2-x+1}$$  

• $$\frac{\frac{4}{3}}{x+1}+\frac{-\frac{1}{3}x+\frac{5}{3}}{x^2+1}$$  

• $$\frac{\frac{4}{3}}{x+1}+\frac{-\frac{5}{3}x+\frac{1}{3}}{x^2-x+1}$$  

Multiple Choice

Write the partial fraction decomposition set up for

$$\frac{x^2+3x-2}{x\left(x^2+2x+5\right)}$$  

• $$\frac{Ax+B}{x^2+2x+5}+\frac{C}{x}$$  

• $$\frac{A}{x^2+2x+5}+\frac{B}{x}$$  

• $$\frac{Ax^2+Bx}{x^2+2x+5}+\frac{C}{x}$$  

Case 4 coming up

repeating Irreducible quadratic factors (denominator)

Multiple Choice

write the initial set up

$$\frac{x^3+x-7}{x^2\left(x^2+1\right)^2}$$  

• $$\frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{\left(x^2+1\right)^2}$$  

• $$\frac{A}{x}+\frac{B}{x}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{\left(x^2+1\right)^2}$$  

• $$\frac{A}{x}+\frac{B}{x}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{\left(x^2+1\right)}$$  

Summary for PFD, all denominators completely factored Case:

• 1: nonrepeat linear factors

• 2: repeating linear factors

• 3: nonrepeating quadratic factors

• 4: repeating quadratic factors

• Go practice, we got this!