Problem 48 - Precalculus with Limits

Question

Improper Rational Expression Decomposition, write the partial fraction decomposition of the improper rational expression. $$\frac{16 x^{4}}{(2 x-1)^{3}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given expression \(\frac{16 x^{4}}{(2 x-1)^{3}}\) is \(8x - 12x^{2} + 18x^{3} + \frac{-24x^{2} + 16x - 8}{(2x-1)^{3}}\).

Step 1: Check if the expression is proper or improper

Check the expression \(\frac{16 x^{4}}{(2 x-1)^{3}}\). The degree of the numerator (4) is greater than the degree of the denominator (3), so it is an improper fraction and long division is needed.

Step 2: Perform Polynomial Long Division

Divide \(16x^{4}\) by \((2x-1)^3\) for converting the given improper fraction into a sum of a polynomial and a proper fraction. The result of the division is \(8x - 12x^{2} + 18x^{3}\).

Step 3: Construct the Partial Fraction Decomposition

Write the function as the sum of the polynomial found in Step 2 and a proper fraction, and express the proper fraction as a sum of simpler fractions. Thus, the partial fraction decomposition is \(8x - 12x^{2} + 18x^{3} + \frac{-24x^{2} + 16x - 8}{(2x-1)^3}\).

Key Concepts

Improper Rational Expressions

In the realm of calculus and algebra, an improper rational expression is essentially a fraction where the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. The degree of a polynomial is the highest power of the variable in the expression. When we encounter an improper rational expression, such as \(\frac{16x^4}{(2x-1)^3}\), it signifies that the top is larger or equal to the bottom.

This is important because standard techniques for simplifying fractions don't apply directly to these forms. Instead, we often resort to polynomial long division to break it down into something easier to work with. Improper fractions often appear when dealing with integrals and solving algebraic equations, requiring us to manipulate them into simpler forms for easier calculation.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to long division with numbers. This technique comes into play whenever we encounter an improper rational expression. Let's consider the example \(\frac{16x^4}{(2x-1)^3}\).

Here are the steps followed:

• Identify the dividend (numerator, \(16x^4\)) and the divisor (denominator, \((2x-1)^3\)).
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by that result and subtract from the dividend.
• Repeat the process using the new dividend obtained from the subtraction.

The purpose is to keep reducing the order of the numerator until what remains can no longer be divided evenly by the divisor. The outcome will be a quotient and a remainder, where the remainder becomes the new proper fraction part.

Proper Fraction

A proper fraction is simpler than its improper counterpart since its structure ensures that the degree of the numerator is less than the degree of the denominator. Once the polynomial long division completes, what's left is a proper fraction ready for further decomposition into partial fractions. This step is vital for simplifying rational expressions to make them more manageable for integration or solving.

For the expression \(\frac{16x^4}{(2x-1)^3}\), after performing long division, the remainder \(\frac{-24x^2 + 16x - 8}{(2x-1)^3}\) is now properly arranged: degree of numerator (less than 3) is smaller than that of the denominator (exactly 3). This allows us to decompose it further, making sure it's expressed as a sum of even simpler fractions.

Degree of Polynomials

The degree of a polynomial is a critical concept in understanding and working with both improper and proper fractions. It refers to the highest power of \(x\) in the polynomial. For instance, in \(16x^4\) and \((2x-1)^3\), the degrees are 4 and 3, respectively. This degree dictates the behavior and the methods we employ to simplify or decompose rational expressions.

Comparing degrees allows us to identify whether a rational expression is improper or proper. In an improper fraction, the numerator's degree is equal to or higher than the denominator's, demanding division before further simplification. Understanding the degree helps us efficiently tackle partial fraction decomposition by guiding us on how to arrange and process each polynomial.