Write the partial fraction decomposition of each rational expression. 4x²+13x-9/x (x − 1)(x+3)

Verified step by step guidance

1. Identify the form of the rational expression. Since the denominator is factored as \(x(x - 1)(x + 3)\), and all factors are linear and distinct, the partial fraction decomposition will have terms of the form \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
2. Set up the equation by expressing the given rational expression as a sum of partial fractions: \(\frac{4x^{2} + 13x - 9}{x(x - 1)(x + 3)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
3. Multiply both sides of the equation by the common denominator \(x(x - 1)(x + 3)\) to clear the denominators: \$4x^{2} + 13x - 9 = A(x - 1)(x + 3) + B x (x + 3) + C x (x - 1)$.
4. Expand the right-hand side by multiplying out each term: - \(A(x - 1)(x + 3) = A(x^{2} + 2x - 3)\) - \(B x (x + 3) = B(x^{2} + 3x)\) - \(C x (x - 1) = C(x^{2} - x)\) Then combine all terms to get a polynomial in standard form.
5. Group like terms (powers of \(x\)) on the right side and equate the coefficients of corresponding powers of \(x\) from both sides to form a system of equations. Solve this system to find the values of \(A\), \(B\), and \(C\).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including factoring and simplifying, is essential before performing partial fraction decomposition.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors. For partial fraction decomposition, the denominator must be factored into linear or irreducible quadratic factors to set up the correct form of the decomposition.