Problem 37 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{x^{3}+x^{2}+2}{\left(x^{2}+2\right)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression is \(x - \left(\frac{1}{x^{2}+2}\right) - \left(\frac{x+2}{(x^{2}+2)^2}\right)\).

Step 1: Check Degree of the Polynomials

Since the degree of the numerator \((x^{3}+x^{2}+2)\) is larger than the degree of the denominator \((x^{2}+2)^{2}\), we need to perform polynomial division.

Step 2: Perform Polynomial Division

We divide the numerator by the denominator to simplify the expression. This results in \(x\) with a remainder of \(x^{2}-2x-2\). Our expression now is \(x+\frac{(x^{2}-2x-2)}{(x^{2}+2)^2}\).

Step 3: Set Up Partial Fractions

To decompose the remaining fraction, we set up two fractions, since the denominator (x² + 2)² is a term squared. So, we write \(x+\frac{(x^{2}-2x-2)}{(x^{2}+2)^2} = A\frac{1}{x^{2}+2}+ B\frac{x+2}{(x^{2}+2)^2}\).

Step 4: Solve for A and B

Multiply both sides of the equation by \(x^{2} + 2\)² . This gives \(x(x^{2} + 2)² + A(x^{2} + 2) + B(x - 2) = x^{2} - 2x - 2\). Now choose sample points for \(x\) that will eliminate \(A\) and \(B\). If we let \(x = 0\), this gives us \(A = -1\). Similarly if we let \(x = -2\), we obtain \(B = -1\).

Step 5: Write Final Decomposition

Writing the partial fractions with the found values gives the final decomposition: \(x - \left(\frac{1}{x^{2}+2}\right) - \left(\frac{x+2}{(x^{2}+2)^2}\right)\).

Key Concepts

Polynomial Division

To understand polynomial division, think of it as the algebraic equivalent to the long division you learned with numbers. The key difference is that instead of numbers, we're dividing polynomials, which are expressions involving variables raised to various powers.

Polynomial division simplifies complex rational expressions and is frequently the first step in partial fraction decomposition. When the degree of the numerator is higher than the degree of the denominator, as in our example exercise \(\frac{x^{3}+x^{2}+2}{\left(x^{2}+2\right)^{2}}\), we start by dividing the numerator by the denominator. This division process helps us rewrite the original expression in a form where the numerator has a lower degree than the denominator, which is essential for partial fraction decomposition.

Here, you'd perform the division similar to how you'd divide numbers, carefully aligning terms of similar degrees and subtracting products of the divisor and quotients. The goal is to find the quotient that will be added to a simpler fraction with the remainder over the original denominator. The result is a combination of a polynomial (the quotient) and a rational expression that's easier to decompose further.

Rational Expressions

Rational expressions resemble fractions, only instead of having numbers in the numerator and denominator, they contain polynomials. As seen in the exercise \(\frac{x^{3}+x^{2}+2}{\left(x^{2}+2\right)^{2}}\), understanding rational expressions is critical for mastering algebraic techniques like partial fraction decomposition.

A rational expression is considered simplified when the numerator and denominator have no common factors and the degree of the numerator is less than the degree of the denominator. If the numerator's degree is higher, as in our example, polynomial division must be used to rewrite it into a more manageable format for decomposition. Once in the proper form, we can dismantle the rational expression into simpler 'partial' fractions, each with its own unique constant or variable in the numerator.

Algebraic Techniques

Algebraic techniques are the various methods used to manipulate and solve algebraic equations and expressions. Partial fraction decomposition, which involves breaking down a complex rational expression into simpler fractions, is one such method. It is particularly useful when integrating or working with Laplace transforms.

In our example, after using polynomial division to simplify the expression, we moved to decompose the rational part into a sum of fractions. This involved setting up an equation with unknowns, such as \(A\) and \(B\), in the numerators. Multiplying through by the common denominator, we get an equation where we can find values for these variables by strategically choosing \(x\) values that simplify the equation.

This process of solving for \(A\) and \(B\) often uses 'zeroing out' techniques by selecting values for \(x\) that make terms disappear. Once \(A\) and \(B\) are found, we insert them back into the partial fractions to obtain the final decomposed form, which in the case of the given example becomes \(x - \frac{1}{x^{2}+2} - \frac{x+2}{(x^{2}+2)^2}\). For increased clarity, we often use step-by-step methods to ensure students can follow and comprehend the solution process.