Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. $$(4x^2 - x - 15)/(x(x + 1)(x - 1))$$

Accepted Answer

Identify the form of the partial fraction decomposition based on the factors in the denominator. Since the denominator is $$x(x + 1)(x - 1)$$, which consists of three distinct linear factors, the decomposition will be of the form: $$\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}$$, where $$A$$, $$B$$, and $$C$$ are constants to be determined. Write the equation equating the original rational expression to the sum of the partial fractions: $$\frac{4x$$^{2}$$ - x - 15}{x(x + 1)(x - 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}. $$Multiply both sides of the equation by the common denominator $$x(x + 1)(x - 1) to $$clear the denominators, resulting in: $$4x^{2} - x - 15 = A(x + 1)(x - 1) + B x (x - 1) + C x (x + 1$$)$$. Expand each term on the right-hand side: 
- A(x + 1)(x - 1$$) = $$A(x^{2} - 1$$)$$,
- B x (x - 1$$) = $$B(x^{2} - x$$)$$,
- C x (x + 1$$) = $$C(x^{2} + x$$)$$.
Then combine like terms to express the right side as a polynomial in x$. Set the coefficients of corresponding powers of x$ on both sides equal to each other to form a system of equations. Solve this system for A$, B$, and C$ to find the constants for the partial fraction decomposition.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x^2 - x - 15)/(x(x + 1)(x - 1))

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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Setting Up and Solving Systems of Equations