Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically. Then assign a value to the constant \(a\) and use a graphing utility to check the result graphically. $$ \frac{1}{(x+1)(a-x)} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational function is \(\frac{1}{(a+1)(x+1)} - \frac{1}{(a+1)(a - x)}\), where \(a\) is any real number.

Step 1: Write Down the Partial Fraction Decomposition

We start by expressing the given rational function as the sum of simpler fractions. Given: \(\frac{1}{(x+1)(a-x)} = \frac{A}{x+1} + \frac{B}{a-x}\). Now our task is to find the values of constants A and B.

Step 2: Solve for the Constant Variables

First, multiply both sides by \((x+1)(a-x)\) to eliminate the denominators: \(1 = A(a - x) + B(x + 1)\). Then, set specific values of x for example take \(x = -1\) and \(x = a\). From \(x = -1\), we get: \(1 = A(a + 1)\), so \(A = \frac{1}{a+1}\). From \(x = a\), we get: \(1 = B(a + 1)\), so \(B = \frac{1}{a+1}\). Thus, \(\frac{1}{(x+1)(a-x)} = \frac{1}{(a+1)(x+1)} - \frac{1}{(a+1)(a - x)}\).

Step 3: Assign a value to 'a' and graph

Let's pick a value for \(a\), for instance \(a = 2\). Therefore, our expression becomes: \(\frac{1}{(x+1)(2-x)} = \frac{1}{3(x + 1)} - \frac{1}{3(2 - x)}\). We can now validate these expressions graphically by comparing them through a graphing utility tool plotting both the original and decomposed expressions to see that they have the same graph.

Key Concepts

Rational Expressions

Rational expressions are fractions that have polynomials in their numerators and denominators. Just like regular fractions, these can be simplified, added, subtracted, multiplied, or divided. For example, the expression \(\frac{1}{(x+1)(a-x)}\) is a rational expression because both its numerator \(1\) and its denominator \((x+1)(a-x)\) are polynomials.

Breaking down complex rational expressions into simpler ones is often necessary for easier manipulation and computation. This is achieved through a process called partial fraction decomposition. In this process, a complicated fraction is rewritten as a sum of simpler fractions whose denominators are factors of the original denominator. This makes subsequent algebraic operations much more manageable.

Consider a complex rational expression:

• It can be represented as a sum of simpler rational expressions.
• This method requires understanding factorization and entails solving for constants that make the equation valid.

By practicing with various rational expressions, you'll get comfortable with decomposition and gain a better grasp of operating within polynomial algebra.

Algebraic Verification

Algebraic verification involves checking the correctness of a partial fraction decomposition by using algebraic identities and operations. Once you've decomposed a rational expression, it's vital to ensure your solution is correct before moving forward.

Here's how you can verify a partial fraction decomposition:

• Start by multiplying both sides by the common denominator to cancel out fractions.
• Verify that the identity holds true after simplification.
• Substitute specific values for variables to solve for necessary constants in the decomposition.

In our example, once we decomposed \(\frac{1}{(x+1)(a-x)}\) into \(\frac{A}{x+1} + \frac{B}{a-x}\), we verified the solution by setting particular values of \(x\) and then solving for \(A\) and \(B\). This resulted in a simpler expression that can be verified algebraically, confirming that the identities match when expanded. Such verification helps avoid errors and confirms your solution's validity.

Graphing Utility

Graphing utilities are valuable tools for visual learners and for validating algebraic work. A graphing utility, such as an online graphing calculator or graphing software, can help visualize the function and its decomposition, ensuring the two representations are equivalent.

In partial fraction decomposition, after finding your simpler equations, you can plot both the original rational expression and its decomposed forms to check if they align. Here's how you can use a graphing utility effectively:

• Input both the original and decomposed expressions into the graphing tool.
• Compare the graphs visually to ensure they overlap, indicating equivalence.
• Adjust parameters or values if needed to verify correctness further.

In situations where algebraic verification may be complex, visualization can provide more confidence in your solution. For instance, assigning a specific value to \(a\), like \(a = 2\), then graphing both the original expression \(\frac{1}{(x+1)(2-x)}\) and its decomposition \(\frac{1}{3(x + 1)} - \frac{1}{3(2 - x)}\), helps confirm that our algebraic manipulation was accurate.