Problem 53 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{4 x^{2}-1}{2 x(x+1)^{2}}$$

Step-by-Step Solution

Answer

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

Step 1: Analyse the expression in view of the Partial Fraction Decomposition

A partial fraction decomposition involves expressing the given rational function as a sum of simpler rational functions. In this case, the expression \(\frac{4 x^{2}-1}{2 x(x+1)^{2}}\) can possibly be split up into simpler fractions of the form \(\frac{A}{x}\) and \(\frac{B}{(x+1)^{2}}\), with A and B as the coefficients to be found.

Step 2: Set up the equation and solve for A and B

Going by the approach described in Step 1, equate the given rational expression to a sum of the simpler rational functions so as \(\frac{4 x^{2}-1}{2 x(x+1)^{2}} = \frac{A}{x} + \frac{B}{(x+1)^{2}}\). Multiply through by \(2x(x+1)^{2}\) to clear the denominator, yielding the equation \(4x^{2}-1 = A(x+1)^2 + B(2x)\). Setting x as -1 and 0, you can solve for A and B.

Step 3: Solve for A and B

For x = -1, you'll get the equation 4-1 = A(0) + B(-2) which solves to B = \(\frac{3}{2}\). For x = 0, the equation will be -1 = A(1) which solves to A = -1.

Step 4: Rewrite the original expression

Finally, write the original expression \(\frac{4 x^{2}-1}{2 x(x+1)^{2}}\), but this time replacing A and B with the solutions found in the previous step, yielding \(\frac{-1}{x} + \frac{\frac{3}{2}}{(x+1)^2}\).

Key Concepts

Rational Expressions

Rational expressions are a form of algebraic fractions, involving ratios of polynomial expressions. Much like fractions in basic arithmetic, their numerators and denominators are made up of algebraic terms. In the context of the exercise we consider, the rational expression is
\[\frac{4x^{2}-1}{2x(x+1)^{2}}\].
These expressions can often appear complex and intimidating, but fear not, as they are simply an extension of the fractions you've already mastered. When you encounter a complicated expression like this, partial fraction decomposition provides an effective strategy for breaking it down into simpler, more manageable pieces.

Polynomial Division

Polynomial division is an algebraic process similar to long division with numbers. When dividing polynomials, one polynomial, called the dividend, is divided by another, the divisor, resulting in a quotient and sometimes a remainder. The aim of this operation in the context of partial fraction decomposition is to simplify the given rational expression into a form that can be easily dissected into its constituent parts. To illustrate, if the numerator of a rational expression has a higher degree than the denominator, polynomial division can be used to reduce it such that the leftover fraction has a numerator with a lower degree than the denominator. Subsequently, once the polynomial division is addressed, the expression can be more readily subjected to partial fraction decomposition.

Algebraic Fractions

Algebraic fractions, another term for rational expressions, operate under the same principles as numerical fractions. They can be simplified, added, subtracted, multiplied, or divided. To handle algebraic fractions efficiently, it's essential to have a solid understanding of factoring polynomials and finding the least common denominator (LCD) for combining fractions. Pertaining to the exercise at hand, part of the goal is to express
\[\frac{4x^{2}-1}{2x(x+1)^{2}}\]
as a sum of simpler algebraic fractions through partial fraction decomposition. This not only makes the expression easier to work with but also becomes particularly valuable when integrating mathematical functions or solving integrals within calculus. By mastering algebraic fractions, one is then equipped to tackle a range of mathematical problems in a structured and simplified manner.