Problem 19 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. \frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}

Step-by-Step Solution

Answer

To find the partial fraction decomposition of the given rational function \(\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}\), we need to find constants A, B, and C such that the given function can be written as \(\frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5}\). After solving the derived equations in step 2, we can insert the found values of A, B, and C into this expression to obtain the equivalent form.

Step 1: Split into Partial Fractions

Split the rational function into three fractions: \\[\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5} \\] where A, B, and C are constants to be found, and set them to match the original rational function.

Step 2: Solve for A,B, and C

Multiply each side of the equation by the denominator of the original rational function to clear the fractions: \\[4x^2 -5x -15 = Ax(x+1)(x-5) + Bx(x+1)(x-5) + Cx(x+1)(x-5)\\] Now simplify and collect like terms then compare coefficients with the original equation, thus obtaining three equations in three unknowns A, B, and C. Solve these equations to find the values of A, B, and C.

Step 3: Substitute A,B, and C

Substitute the values of A, B, and C that were obtained in Step 2 into the partial fraction decomposition: \\[\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5}\\] Thus, we have completed the Partial Fraction Decomposition.

Key Concepts

Rational Expressions

A rational expression, in mathematics, is similar to a fraction, but instead of integers, you have polynomials in the numerator and the denominator. In simple terms, it’s a fraction that has a polynomial at the top and bottom. The rational expression in our original exercise is \(\frac{4x^2-5x-15}{x(x+1)(x-5)}\), which consists of a polynomial of degree 2 in the numerator and the product of three linear factors in the denominator.

Understanding the composition of rational expressions is critical. Simplifying and working with these expressions requires factoring polynomials, canceling common factors, and knowing how to find a least common denominator when adding or subtracting. For instance, the denominator of the given expression factors into three different linear expressions, making it a prime candidate for partial fraction decomposition. This process breaks down complex rational expressions into simpler ones, which are easier to integrate, differentiate, or simply understand.

Algebraic Fractions

Algebraic fractions are the fractions wherein the numerator or denominator (or both) contain algebraic expressions. In our exercise, the algebraic fraction is complex due to its detailed numerator and a product of linear terms in the denominator. The idea behind dealing with algebraic fractions is to simplify them, if possible, or break them down into simpler parts, known as partial fractions.

Steps to Simplify Algebraic Fractions

• Factor the numerator and the denominator completely.
• Divide out any common factors between the numerator and the denominator.
• Rewrite or decompose complex fractions into simpler parts if necessary.

Partial fraction decomposition is a particularly useful technique when it comes to integration in calculus, so simplifying algebraic fractions is a foundational skill beneficial across different areas of mathematics.

Polynomial Equations

Polynomial equations are mathematical expressions formed by equating a polynomial to a value, often zero. A polynomial is composed of variables and coefficients, utilizing only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, in step 2 of the given solution, we see a polynomial equation when the terms are collected on one side.

By setting up the equation \(4x^2 -5x -15 = Ax(x+1)(x-5) + Bx(x+1)(x-5) + Cx(x+1)(x-5)\), we are asserting that the original rational expression can be represented as a sum of simpler fractions. The polynomial on the right side, when expanded and simplified, must then equate to the polynomial on the left side. Solving for the coefficients A, B, and C involves equating the coefficients of like terms (terms with the same powers of x) on both sides of the equation, which is a fundamental approach when working with polynomial equations in algebra.