Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients. (a) $\frac{3x}{(x + 3)(3x + 4)}$, (b) $\frac{4}{x^3 + 6x^2 +9x}$

Neetesh Kumar | December 9, 2024

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This is the solution to Math 132
Assignment: 7.4 Question Number 2
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Step-by-step solution:

(a) $\frac{3x}{(x+3)(3x+4)}$

Solution:

The denominator $(x+3)(3x+4)$ consists of two distinct linear factors.

The partial fraction decomposition is:

$\frac{3x}{(x+3)(3x+4)} = \frac{A}{x+3} + \frac{B}{3x+4}$.

(b) $\frac{4}{x^3 + 6x^2 + 9x}$

Solution:

The denominator $x^3 + 6x^2 + 9x$ factors as $x(x+3)^2$.

The partial fraction decomposition is:

$\frac{4}{x^3 + 6x^2 + 9x} = \frac{A}{x} + \frac{B}{x+3} + \frac{C}{(x+3)^2}$.

Final Answers:

(a) $\frac{3x}{(x+3)(3x+4)} = \frac{A}{x+3} + \frac{B}{3x+4}$

(b) $\frac{4}{x^3 + 6x^2 + 9x} = \frac{A}{x} + \frac{B}{x+3} + \frac{C}{(x+3)^2}$

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