Problem 67 - 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. $$\frac{4 x^{2}+2 x-1}{x^{2}(x+1)}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Form of the Decomposition

We will first factor the denominator \(x^{2}(x+1)\). After factoring, we see that the denominator consists of two different factors, \(x\) (with multiplicity 2) and \(x+1\). Therefore, the expression will decompose in the form \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+1}\).

Step 2: Solve for A, B and C

We multiply through by the common denominator to remove the fractions and get the following equation: \(4x^{2}+2x-1=A(x)(x+1)+B(x+1)+Cx^{2}\). Now, we equate the coefficients: On comparing coefficients of \(x^{2}\): \(4=C\), comparing coefficients of \(x\): \(2=A+B\), and comparing constants term: \(-1=B\). Substituting \(B=-1\) in \(A+B=2\), we get \(A=3\).

Step 3: Write the Decomposition with the Found Coefficients

Now that we've found the values for A, B, and C, we can rewrite the original expression as \(\frac{3}{x} - \frac{1}{x^{2}} + \frac{4}{x+1}\) .

Step 4: Validate the Result

We can check our result algebraically by taking the common denominator and adding the fractions. We can also use a graphing calculator to check our result graphically. Graph both the original expression and the result to see if they are indeed the same. If they are, this validates our result.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. The expression \(\frac{4x^2 + 2x - 1}{x^2(x+1)}\) is a rational expression since both parts are polynomials. Understanding these is crucial as they appear frequently in calculus, algebra, and other areas of mathematics.

Rational expressions can be simplified or manipulated using various techniques, one of which is partial fraction decomposition. This is particularly useful when integrating rational expressions or when finding their limits. It involves breaking down the complex rational expression into a sum of simpler fractions, whose denominators are factors of the original denominator.

Students should get comfortable with handling the algebra of these expressions, like finding a common denominator to combine them, or splitting them into separate terms that can be managed more easily. Knowing these basics will make the more advanced operations much more approachable.

Factoring Polynomials

Factoring polynomials is a process of breaking down a polynomial into products of its factors that, when multiplied together, give back the original polynomial. It's a foundational tool for handling rational expressions. In our exercise, the polynomial \(x^2(x+1)\) is already factored. However, recognizing when and how to factor polynomials is key in the process of partial fraction decomposition.

Why is factoring important?

By factoring, we can easily simplify expressions, solve equations, and find zeros of functions. In the case of partial fraction decomposition, factoring allows us to split the denominator into parts which we can then handle individually. There are various methods of factoring polynomials, like finding a common factor, using the difference of squares, or applying the quadratic formula when necessary.

For students struggling with factoring, it is advisable to practice this skill separately. Start with simple polynomials and gradually move to more complex ones. Building this foundation will immensely help with understanding and executing partial fraction decomposition.

Graphing Calculator Validation

With advancements in technology, graphing calculators have become an indispensable tool for validating solutions in mathematics. After performing a partial fraction decomposition, it is highly recommended to check your work using a graphing calculator. For instance, you can compare the graph of the original rational expression \(\frac{4x^2 + 2x - 1}{x^2(x+1)}\) with the graph of the decomposed expression \(\frac{3}{x} - \frac{1}{x^2} + \frac{4}{x+1}\).

By graphing both expressions in the same viewing window, students should see that the graphs overlap completely, indicating a correct decomposition. This visual confirmation is an excellent way to ensure accuracy. Students should note, though, that graphing utilities have limitations and may not display functions accurately at all scales, so it's also important to check the work algebraically to guarantee that the graphing result is reliable.

Graphing calculators shouldn't be used as a shortcut to avoid understanding the underlying concepts, but rather as a tool for confirming an already grounded understanding. They can also be used to explore and visualize concepts, which can further solidify the student's comprehension of rational expressions and their behaviors.