Problem 41 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{8 x-12}{x^{2}\left(x^{2}+2\right)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational function is \( \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{2}+2} + \frac{Dx + E}{(x^{2} + 2)^{2}} \), where constants A, B, C, D, E are the values determined in the steps above.

Step 1: Identify the general form

For the given rational function \(\frac{8 x-12}{x^{2}(x^{2}+2)^{2}}\), identify its general form of partial fraction decomposition which is given by \( \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{2}+2} + \frac{Dx + E}{(x^{2} + 2)^{2}} \)

Step 2: Equate the rational function to the general form

Equate the given rational function to the general form to match both sides, i.e, \(\frac{8 x-12}{x^{2}(x^{2}+2)^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{2}+2} + \frac{Dx + E}{(x^{2} + 2)^{2}} \). Now, clear the fraction by multiplying both sides by \(x^{2}(x^{2}+2)^{2}\)

Step 3: Solve for the constants

After clearing fractions, we get \(8x - 12 = Ax(x^{2} + 2)^{2} + B(x^{2} + 2)^{2} + Cx(x^{2} + 2) + Dx^{2} + Ex\). It's common to choose convenient values for \(x\) to make most of the terms zero and easy to solve. Let's use \(x = 0\) and \(x = \sqrt{2}\) to solve for constants \(B, A, D, E\) and \(C\) respectively.

Step 4: Check the result

Once the constants are found, substitute them back into the general form to obtain the partial fraction decomposition. Then, add these simpler fractions and verify that the result is the same as the given rational function.

Key Concepts

Rational Expression

A rational expression is simply a fraction where both the numerator and the denominator are polynomials. In the context of partial fraction decomposition, you're usually dealing with complex rational expressions. For instance, consider the expression \(\frac{8x-12}{x^{2}(x^{2}+2)^{2}}\). This looks complicated at first glance, but it's just a division of one polynomial by another.
To simplify or manipulate such expressions, it's often helpful to break them down into simpler pieces. This is exactly what partial fraction decomposition achieves. The main goal is to express a complex rational expression as a sum of simpler fractions, which are easier to integrate or work with in algebraic equations.

Algebraic Verification

Algebraic verification involves ensuring that the partial fraction decomposition you've found is correct. This means that when you recombine your simpler fractions, they should equal the original rational expression.
In practice, after assigning constants to each fraction, you solve equations to determine their values. Finally, by substituting these constants back into the decomposed form, you rebuild the complex fraction. If it matches the initial expression, the decomposition is verified as correct.
Verification steps often use identity properties or plug values for variables that simplify the equation. These techniques confirm that every term aligns with the decomposition and no mistakes were made.

General Form

The general form in partial fraction decomposition refers to the standard way in which a rational expression is broken down into simpler fractions.
The process begins by identifying distinct polynomial factors and assigning each a type of fraction. For instance, our example \(\frac{8x-12}{x^{2}(x^{2}+2)^{2}}\) can be decomposed into \(\frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{2}+2} + \frac{Dx + E}{(x^{2} + 2)^{2}}\).
This form is crucial as it sets the framework for determining the unknown constants \(A, B, C, D,\) and \(E\). By following this standardized approach, any partial fraction decomposition becomes more systematic and less error-prone.

Solve for Constants

Solving for constants in partial fraction decomposition involves algebraic manipulation to find the values of \(A, B, C, D,\) and \(E\). After expressing the rational expression in its general form, equate it to the original rational function.
The next step is to clear the fractions by multiplying both sides by the common denominator. You can then choose strategic values for \(x\) that eliminate most terms, hence simplifying calculations.

• For example, by choosing \(x = 0\), certain terms drop out, making it easier to solve for \(B\).
• Choosing \(x = \sqrt{2}\) may isolate terms that let you solve for \(C, D,\) or \(E\).

Through these clever substitutions, solving these constants becomes a manageable task. Finally, ensure their correctness by substituting back into the decomposed form and verifying with the original expression.