Answer:
[tex]\frac{3}{x + 3} + \frac{2}{(x+3)^2}[/tex]
Step-by-step explanation:
Given
[tex]\frac{3x + 11}{x^2 +6x + 9}[/tex]
Required
Express as partial fraction
[tex]\frac{3x + 11}{x^2 +6x + 9}[/tex]
Expand the numerator
[tex]\frac{3x + 11}{x^2 +3x +3x+ 9}[/tex]
Factorize
[tex]\frac{3x + 11}{x(x +3) +3(x+ 3)}[/tex]
Factor out x + 3
[tex]\frac{3x + 11}{(x +3)(x+ 3)}[/tex]
[tex]\frac{3x + 11}{(x +3)^2}[/tex]
As a partial fraction, we have:
[tex]\frac{3x + 11}{(x +3)^2} = \frac{A}{x + 3} + \frac{B}{(x+3)^2}[/tex]
Take LCM
[tex]\frac{3x + 11}{(x +3)^2} = \frac{A(x+3) + B}{(x + 3)^2}[/tex]
Cancel out (x + 3)^2 on both sides
[tex]3x + 11 = A(x+3) + B[/tex]
Open bracket
[tex]3x + 11 = Ax+3A + B[/tex]
By comparison, we have:
[tex]Ax = 3x[/tex] ===> [tex]A = 3[/tex]
[tex]3A + B = 11[/tex]
Substitute 3 for A
[tex]3*3 + B = 11[/tex]
[tex]9 + B = 11[/tex]
Solve for B
[tex]B = 11-9[/tex]
[tex]B =2[/tex]
Substitute: [tex]A = 3[/tex] and [tex]B =2[/tex] in
[tex]\frac{3x + 11}{(x +3)^2} = \frac{A}{x + 3} + \frac{B}{(x+3)^2}[/tex]
[tex]\frac{3x + 11}{(x +3)^2} = \frac{3}{x + 3} + \frac{2}{(x+3)^2}[/tex]
Hence, the partial fraction is:
[tex]\frac{3}{x + 3} + \frac{2}{(x+3)^2}[/tex]
