Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{2}+12 x-9}{x^{3}-9 x}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the expression \(\frac{x^{2}+12 x-9}{x(x-3)(x+3)}\) is \(\frac{1}{x} + \frac{3}{x-3} - \frac{4}{x+3}\)

Step 1: Factorize and simplify the expression

The expression can be rewritten, factoring the denominator to \(x^{2}+12 x-9 = (x+9)(x-1)\) and \(x^{3}-9 x = x(x^{2}-9) = x(x-3)(x+3)\). So, our modified expression becomes \(\frac{x^{2}+12 x-9}{x(x-3)(x+3)}\)

Step 2: Write the expression as a sum of partial fractions

The partial fraction decomposition of the expression will have the form \(\frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\). Multiply both sides by the denominator of the left side: \(x^{2}+12 x-9 = A(x-3)(x+3) + Bx(x+3) + Cx(x-3)\). Expanding and comparing coefficients, we obtain a system of equations: \n1) For \(x^{2}\) coefficients: 0 = A + B + C,\n2) For \(x\) coefficients: 12 = -3A + 3B - 3C, \nAnd taking x = 0 we get: -9 = -9A.\nSolving this system we get A = 1, B = 3, C = -4.

Step 3: Combine the fractions and check the result

Substituting A, B, and C back into the partial fraction, \(\frac{1}{x} + \frac{3}{x-3} - \frac{4}{x+3}\). To check the answer, combine these fractions and simplify. Multiply everything by \(x(x-3)(x+3)\) to clear the denominators. This should give us back our original expression: \(x^{2}+12 x-9\). Similarly, the graphical check involves superimposing the graph of the initial expression over the graph of the combined partial fractions to see if they match, indicating a correct decomposition.

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In the context of our exercise, the rational expression is \(\frac{x^2 + 12x - 9}{x^3 - 9x}\).Such expressions often appear in calculus and algebra and can represent complex relationships between variables. A key aspect of working with rational expressions is simplifying them, which can involve factoring and reducing fractions.

When simplifying rational expressions, it's important to remember that like regular fractions, you can only reduce them when the numerator and the denominator have common factors. In the provided example, we are asked to perform 'partial fraction decomposition', which is a technique used to break down a complex rational expression into simpler parts called 'partial fractions'. This process is particularly useful when integrating rational expressions.

If we can express our given rational expression as a sum of simpler fractions, we can more easily understand the behavior of the graph, identify vertical asymptotes, holes, or intercepts, and perform more advanced calculus operations like integration. Graphically checking the partial fractions involves plotting the original expression and the decomposed expression on the same graph to see if they coincide, validating our algebraic manipulation.

The Role of Factoring Polynomials

Factoring polynomials stands at the heart of many algebraic processes, including partial fraction decomposition. It's a method of breaking down a polynomial into the product of its factors, which are polynomials of lower degree. In our example, \(x^3 - 9x\) is factored into \(x(x - 3)(x + 3)\), which simplifies the rational expression in a form suitable for partial fraction decomposition.

Why is factoring important? Factoring gives us insights into the roots of the polynomials, those values of x for which the polynomial equals zero. These roots are key in solving equations and inequalities. They also indicate the x-values where the graph of the polynomial will intersect the x-axis. In partial fraction decomposition, knowing the factors helps us determine the denominators of the partial fractions and their possible numerators.

To factor a polynomial, one might look for a common factor, apply the difference of squares or other recognizable patterns, or use techniques such as grouping. The ability to factor polynomials effectively improves not only your algebraic manipulation skills but also your understanding of mathematical functions and graphs.

Solving Systems of Equations

Systems of equations are collections of equations that you solve simultaneously. They represent multiple constraints on a set of variables that must be satisfied at the same time. In the context of partial fraction decomposition, once we set up our equation with the unknown coefficients (A, B, C), we end up with a system of equations to solve.

In our example, the system we derive from equating coefficients is:

• For \(x^2\) coefficients: 0 = A + B + C,
• For \(x\) coefficients: 12 = -3A + 3B - 3C,
• And taking x = 0 we get: -9 = -9A.

To solve this system, we can use methods like substitution, elimination, or even matrix operations for larger systems. Solving for A, B, and C allows us to complete the decomposition process. Understanding how to solve these equations is crucial when dealing with partial fractions because it allows us to find the coefficients that split our complex rational expression into simpler, more manageable parts.

Mastering the process of solving systems of equations not only aids in tasks like partial fraction decomposition but also in various fields where multiple conditions must be met simultaneously, such as in economics, physics, and computer science.