Question

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

Accepted Answer

Identify the form of the partial fraction decomposition. Since the denominator is $$(x + 4)(3x^2 + 1)$$, where $$x + 4 is a $$linear factor and $$3x^2 + 1 is an $$irreducible quadratic factor, the decomposition will be of the form: $$ \frac{3x - 2}{(x + 4)(3x^2 + 1)} = \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1} $$ where $$A$$, $$B$$, and $$C$$ are constants to be determined. Multiply both sides of the equation by the common denominator $$(x + 4)(3x^2 + 1) to $$clear the fractions: $$ 3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4) $$ This step eliminates the denominators and allows us to work with polynomials. Expand the right-hand side by distributing: $$ A(3x^2 + 1) = 3A x^2 + A $$ and $$ (Bx + C)(x + 4) = Bx^2 + 4Bx + Cx + 4C . $$Combine like terms to write the right side as a polynomial in standard form: $$ (3A + B) x^2 + (4B + C) x + (A + 4C) . $$Set up a system of equations by equating the coefficients of corresponding powers of $$x$$ from both sides. On the left, the polynomial is $$3x - 2$$, which can be written as $$0x^2 + 3x - 2. So$$, equate coefficients: $$ 	ext{Coefficient of } x^2: 0 = 3A + B $$ $$ 	ext{Coefficient of } x: 3 = 4B + C $$ $$ 	ext{Constant term}: -2 = A + 4C . $$Solve the system of equations for $$A$$, $$B$$, and $$C. $$Once these constants are found, substitute them back into the partial fraction form $$ \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1}  to $$complete the decomposition.

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

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Key Concepts

Partial Fraction Decomposition

Factoring and Types of Denominator Factors

Setting Up and Solving Systems of Equations