Problem 26 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{x+2}{x\left(x^{2}-9\right)}$$

Step-by-Step Solution

Answer

After working through the steps, the constants A, B, and C and hence the partial fraction decomposition can be found. The correct decomposition should be validated by checking if it simplifies to the original fraction.

Step 1: Factorize the denominator

The first step is to factorize the denominator \(x(x^{2}-9)\). This can be further factored into \(x(x-3)(x+3)\), as \(x^{2}-9\) can be recognized as a difference of squares, hence \(x(x-3)(x+3)\).

Step 2: Set up partial fractions

Set up the partial fractions. Each factor in the denominator will correspond to a term in the partial fraction decomposition. Hence, express \(\frac{x+2}{x(x-3)(x+3)}\) as \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\), where A, B, and C are constants to be determined.

Step 3: Equate expressions and solve for A, B, and C

Next, we multiply both sides by the common denominator \(x(x-3)(x+3)\) to clear out the fractions. It follows that: \(x+2 = A(x-3)(x+3) + Bx(x+3) +Cx(x-3)\). Now we look for the values of A, B, and C by setting up and solving a system of equations. We can derive these equations by substituting simple numbers for \(x\) or by comparing coefficients on each side.

Step 4: Validate the result

In the end, substitute the found values of A, B, and C back into the partial fractions \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\) and simplify. If the result is equal to the original fraction, the partial fraction decomposition is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra, and it lays the foundation for many concepts like partial fraction decomposition. When you look at a polynomial, you want to break it down into simpler pieces or factors that, when multiplied together, give you the original polynomial. This can simplify working with polynomials significantly. A common technique is to look for patterns such as the difference of squares, which was used in the exercise. The expression \(x^2-9\) is a classical example, as it can be written as \((x-3)(x+3)\). This recognition allows us to transform the original polynomial into its factored form, making it easier to work with. By factoring \(x(x^2-9)\) into \(x(x-3)(x+3)\), we simplify the problem by identifying each unique factor.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding them is crucial because they appear frequently in algebraic equations and need to be simplified or decomposed in partial fraction problems. When we are given a rational expression, such as \(\frac{x+2}{x(x-3)(x+3)}\), we first check if we can simplify it further by factoring. However, in this case, the expression is already factored, allowing us to set up for the partial fraction decomposition directly. The aim is often to express more complex rational expressions in simpler forms, which can be achieved by breaking it down into a sum of simpler fractions. This step is vital when solving integrals or comparing different functions.

Algebraic Equations

Algebraic equations are equations formed by polynomials and can include rational expressions. Solving these equations often involves multiple steps and different strategies. In the context of partial fraction decomposition, these strategies come into play when setting up and solving for unknowns. To achieve the partial fraction decomposition, you start by equating the original expression with the sum of its tentative partial fractions, like \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\). You then use algebraic equations to find the constants \(A\), \(B\), and \(C\). This requires solving a system of equations: setting different values for \(x\) or comparing coefficients to find unknowns. Finally, you confirm your result by substituting back, ensuring the original expression is retained. Mastering algebraic equations is crucial as it allows you to tackle problems methodically and with confidence.