6.3 Partial Fractions

If the degree of numerator P(x) is equal or greater than the degree of denominator D(x), then the fraction is:

• Proper

• Improper

• Both proper and improper

• None of the above

If the degree of numerator P(x) is less than the degree of denominator D(x), then the fraction is:

• Proper

• Improper

• Both proper and improper

• None of the above

 This fraction is known as $\frac{2x+5}{x^2+5x+6}$ .

• Proper

• Improper

• Both proper and improper

• None of the above

Which one should be used in the partial fraction decomposition of   $\frac{2x^2-x+4}{x^2-x-2}$  ?

•  $2+\frac{-3x}{x^2-x-2}$  

•  $2+\frac{3x}{x^2-x-2}$  

•  $2+\frac{x+8}{x^2-x-2}$  

• None, the fraction is proper.

Which one should be used in the partial fraction decomposition of  $\frac{4x}{\left(x+1\right)\left(3+x^2\right)}$  ?

•  $\frac{A}{(x+1)​}+\frac{B}{(3+x)}​+\frac{C​}{\left(3+x^2\right)}$  

•  $\frac{A}{(x+1)}​+\frac{Bx+C​}{\left(3+x^2\right)}$  

•  $\frac{A}{(x+1)}​+\frac{B​}{\left(3+x^2\right)}$  

•  $\frac{A}{(x+1)}​+\frac{Bx​}{\left(3+x^2\right)}$  

Which one should be used in the partial fraction decomposition of  $\frac{x-11}{\left(x+3\right)^2\left(x-4\right)}$  ?

•  $\frac{A}{(x+3)​}+\frac{B}{(x+3)}​+\frac{C​}{(x-4)}$  

• 
   $\frac{A}{\left(x-4\right)}​+\frac{Bx+C​}{(3+x)^2}$  

•  $\frac{A}{(x+3)^2}​+\frac{B}{(x-4)}$  

•  $\frac{A}{(x+3)}​+\frac{B}{(x+3)^2}​+\frac{C​}{(x-4)}$  

Given $\frac{x-2}{\left(x-3\right)\left(2x+1\right)}=\frac{A}{x-3}+\frac{B}{2x+1}$ . Find A and B.

•  $\frac{1}{7}\ and\ \frac{5}{7}$  

•  $-\frac{1}{7}\ and\ -\frac{5}{7}$  
  

•  $-\frac{1}{7}\ and\ \frac{5}{7}$  
  

•  $\frac{1}{7}\ and\ -\frac{5}{7}$  
  

Given $\frac{7-x}{\left(x-1\right)\left(2x+1\right)}=\frac{A}{x-1}+\frac{B}{2x+1}$ . Find A and B.

•  $2\ and\ 5$  

•  $-5\ and\ 2$  
  

•  $2\ and\ -5$  
  

•  $-2\ and\ 5$  
  

Find the partial fraction decomposition $\frac{-5x-41}{x^2+x-12}$  .

•  $\frac{3x}{x+4}-\frac{8x}{x-3}$  

•  $\frac{3}{x+4}-\frac{8}{x-3}$  

•  $-\frac{21}{x-3}+\frac{16}{x+4}$  

•  $-\frac{8}{x+4}+\frac{3}{x-3}$  

Express $\frac{2x^3-8x^2+13x-17}{2x^2+x-6}$ in partial fractions. 

•  $x-\frac{9}{2}-\frac{5}{2\left(2x-3\right)}+\frac{13}{x+2}$  

•  $x-\frac{9}{2}-\frac{5}{2\left(2x-3\right)}-\frac{13}{x+2}$  

•  $x+\frac{9}{2}+\frac{5}{2\left(2x-3\right)}+\frac{13}{x+2}$  

•  $x-\frac{9}{2}+\frac{5}{2\left(2x-3\right)}+\frac{13}{x+2}$