Question

Ezpress 3x + 11/x² + 6x + 9as partial fractions,​

Answer 1

Answer:

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

Step-by-step explanation:

Given

$\frac{3x + 11}{x^2 +6x + 9}$

Required

Express as partial fraction

$\frac{3x + 11}{x^2 +6x + 9}$

Expand the numerator

$\frac{3x + 11}{x^2 +3x +3x+ 9}$

Factorize

$\frac{3x + 11}{x(x +3) +3(x+ 3)}$

Factor out x + 3

$\frac{3x + 11}{(x +3)(x+ 3)}$

$\frac{3x + 11}{(x +3)^2}$

As a partial fraction, we have:

$\frac{3x + 11}{(x +3)^2} = \frac{A}{x + 3} + \frac{B}{(x+3)^2}$

Take LCM

$\frac{3x + 11}{(x +3)^2} = \frac{A(x+3) + B}{(x + 3)^2}$

Cancel out (x + 3)^2 on both sides

$3x + 11 = A(x+3) + B$

Open bracket

$3x + 11 = Ax+3A + B$

By comparison, we have:

$Ax = 3x$ ===> $A = 3$

$3A + B = 11$

Substitute 3 for A

$3*3 + B = 11$

$9 + B = 11$

Solve for B

$B = 11-9$

$B =2$

Substitute: $A = 3$ and $B =2$ in

$\frac{3x + 11}{(x +3)^2} = \frac{A}{x + 3} + \frac{B}{(x+3)^2}$

$\frac{3x + 11}{(x +3)^2} = \frac{3}{x + 3} + \frac{2}{(x+3)^2}$

Hence, the partial fraction is:

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