Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x2 + 4))

Accepted Answer

Identify the denominator factors of the rational expression $$\frac{1}{x(2x + 1)(3x^2$ + 4)}. $$Here, the factors are $$x$$, $$2x + 1$$, and $$3x^2 + 4$. Note that x$ and 2x + 1$ are linear factors, while 3x^2$ + 4 is an $$irreducible quadratic factor. Set up the partial fraction decomposition form. For each linear factor, assign a constant numerator, and for the irreducible quadratic factor, assign a linear numerator. So, write the decomposition as:

$$\frac{1}{x(2x + 1)(3x^2$ + 4)} = \frac{A}{x} + \frac{B}{2x + 1} + \frac{Cx + D}{3x^2$ + 4}$$,

where $$A$$, $$B$$, $$C$$, and $$D$$ are constants to be determined. Multiply both sides of the equation by the common denominator $$x(2x + 1)(3x^2$ + 4) to $$clear the denominators. This gives:

$$1 = A(2x + 1)(3x^2$ + 4) + B x (3x^2$ + 4) + (Cx + D) x (2x + 1). $$Expand the right-hand side by distributing each term carefully. This involves multiplying polynomials and combining like terms to express the right side as a polynomial in powers of $$x. $$Equate the coefficients of corresponding powers of $$x on $$both sides of the equation. This will give a system of linear equations in terms of $$A$$, $$B$$, $$C$$, and $$D. $$Solve this system to find the values of these constants, completing the partial fraction decomposition.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x² + 4))

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Key Concepts

Partial Fraction Decomposition

Types of Factors in the Denominator

Solving for Unknown Coefficients