Problem 72 - 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. $$\frac{4 x^{4}}{(2 x-1)^{3}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{4x^4}{(2x-1)^3}\) is \(\frac{3/4}{2x-1} + \frac{0}{(2x-1)^2} + \frac{1/2}{(2x-1)^3}

Step 1: Identify the Fractions and Form a General Statement

The fraction is already an improper fraction, so we don't have to do a long division. The denominator \((2x-1)^3\) has the common factor \(2x-1\) repeating three times. So, we can form a general statement:\[\frac{4x^4}{(2x-1)^3} = \frac{A}{2x-1} + \frac{B}{(2x-1)^2} + \frac{C}{(2x-1)^3}\]where A, B, and C are constants that we need to find.

Step 2: Clear the Denominator

We multiply through by \((2x - 1)^3\) to clear the denominator:\[4x^4 = A(2x - 1)^2 + B(2x - 1)+ C\]

Step 3: Substitute Suitable Values for x

Now, we choose suitable values for x to eliminate A, B, and C in turn. A suitable value would be x = 1/2, giving:\[4(1/2)^4 = A(0) + B(0) + C\]which simplifies to\[\frac{1}{2} = C\]Now that we have C, we need to find A and B.

Step 4: Differentiate to Find Further Coefficients

Differentiating both sides of the equation (using chain rule for A term) gives:\[16x^3 = 2A(2x-1)(2) + B\]And plugging in the value x = 1/2 again gives B = 0. Now we differentiate this equation again, obtaining: \[48x^2 = 4A \]And using x = 1/2, we get:\[3 = 4A\]Therefore, \(A = 3/4\).

Step 5: Write the Decomposed Fraction

Substitute the calculated values of A, B, and C back into the general decomposition we originally declared. This gives:\[\frac{4x^4}{(2x-1)^3} = \frac{3/4}{2x-1} + \frac{0}{(2x-1)^2} + \frac{1/2}{(2x-1)^3}\]

Key Concepts

Rational Expression Decomposition

When dealing with rational expressions, especially complex ones with higher degree polynomials, we often turn to partial fraction decomposition. This method breaks down a complicated fraction into simpler 'partial' fractions that are easier to integrate, differentiate, or otherwise work with.

Consider the given exercise \[\frac{4 x^{4}}{(2 x-1)^{3}}\]. Our goal is to express this as a sum of simpler rational expressions. We anticipate a decomposition involving terms of the form \[\frac{A}{2x-1}, \frac{B}{(2x-1)^2},\] and \[\frac{C}{(2x-1)^3}\], where A, B, and C are constants to be determined. This decomposition is based on the fact that \[(2x-1)^3\] can be thought of as a product of same factors \[(2x-1)(2x-1)(2x-1)\].

• First, we equate the original rational expression with our proposed partial fractions.
• Next, we clear the denominators to derive a polynomial equation.
• Then, we solve this equation for A, B, and C by strategically choosing values for x or using differentiation.
• Once these constants are found, we substitute them back to get our decomposed expression.

This process simplifies the original expression and is part of a foundation for knowledge in calculus and higher mathematics.

Algebraic Techniques

The exercise employs several algebraic techniques that are crucial for solving a range of mathematical problems. These methods include manipulating equations, differentiating to find coefficients, and strategically substituting values for variables to simplify expressions.

We use these techniques in the solution as follows:

• Multiplication of both sides by the common denominator to eliminate fractions.
• Substitution of specific values of x to simplify the equation and solve for the unknown constants one at a time.
• Differentiation, which takes advantage of the repeated differentiation rule (chain rule) to find coefficients by reducing the degree of the terms.

These methods do more than just help solve for constants in partial fraction decomposition; they are essential for understanding and solving a variety of algebraic and calculus problems. By mastering algebraic techniques, students can tackle a wide range of mathematical challenges with confidence.

Graphing Utility

Using a graphing utility is an exceedingly helpful way to visualize the problem and verify solutions in algebra and calculus. Graphing the original rational expression alongside its decomposed partial fractions serves as a check; both graphs should be identical since they represent the same mathematical relationship.

In our example, we could graph \[\frac{4 x^{4}}{(2 x-1)^{3}}\] using a graphing calculator or software, and then graph \[\frac{3/4}{2x-1} + \frac{0}{(2x-1)^2} + \frac{1/2}{(2x-1)^3}\] separately. If the decomposed fractions have been found correctly, the two graphs will lie on top of one another.

• Graphing the expressions provides a visual confirmation of the algebraic work.
• It can also help identify errors or discrepancies in the algebraic solution.
• Furthermore, visualization aids in understanding the behavior of the function, such as asymptotes and intercepts.

Incorporating graphing utilities into problem-solving not only aids in confirmation but also strengthens comprehension of the underlying mathematical concepts.