Write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

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1. Identify the form of the partial fraction decomposition for the given rational expression. Since the denominator is factored into two distinct linear factors, \((x - 9)\) and \((x + 2)\), the decomposition will be of the form: \(\frac{A}{x - 9} + \frac{B}{x + 2}\), where \(A\) and \(B\) are constants to be determined.
2. Write the equation by setting the original rational expression equal to the sum of the partial fractions: \(\frac{3x + 50}{(x - 9)(x + 2)} = \frac{A}{x - 9} + \frac{B}{x + 2}\).
3. Multiply both sides of the equation by the common denominator \((x - 9)(x + 2)\) to clear the denominators: \$3x + 50 = A(x + 2) + B(x - 9)$.
4. Expand the right side: \$3x + 50 = A x + 2A + B x - 9B\(, then combine like terms: \)3x + 50 = (A + B) x + (2A - 9B)$.
5. Set up a system of equations by equating the coefficients of corresponding terms on both sides: For the \(x\) terms, \$3 = A + B\(; for the constant terms, \)50 = 2A - 9B\(. Solve this system to find the values of \)A\( and \)B$.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with linear or irreducible quadratic denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring polynomials means rewriting them as a product of simpler polynomials. Recognizing factors in the denominator, such as (x - 9)(x + 2), is crucial for setting up the correct form of partial fractions.