Problem 51 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{5-x}{2 x^{2}+x-1}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression is \(\frac{6}{2x - 1} - \frac{5}{x + 1}\).

Step 1: Factor the Denominator

First, we factor the denominator of the rational function. In this case, the denominator \(2x^{2}+x-1\) can be factored into \((2x - 1)(x + 1)\). Thus, our rational expression becomes \(\frac{5-x}{(2x - 1)(x + 1)}\).

Step 2: Setup the Decomposition Structure

Now we set up the decomposition structure of the rational expression. The factorized denominator \((2x - 1)(x + 1)\) gives two linear factors. For each linear factor \(ax + b\), we use \(\frac{A}{ax+b}\) in the decomposition. Here, \(A\) is a constant which we need to find. Thus, the partial fraction decomposition can be written as \[ \frac{5-x}{(2x - 1)(x + 1)} = \frac{A}{2x - 1} + \frac{B}{x + 1} \] where A and B are constants to be determined.

Step 3: Equate and Solve for Constants

We now multiply through by the denominator of the left side to eliminate fractions: \[ 5 - x = A(x + 1) + B(2x - 1) \] From here, we need to find the values of A and B that make this equation true for all x. We can solve for A and B by letting x be the values that make each factor 0, given that those values don't make the function undefined. Let x be -1, then we get \[ 5 - (-1) = A((-1) + 1) + B(2(-1) - 1) \] On simplifying, we get A = 6. Solve for B by letting x = \(\frac{1}{2}\), we get \[ 5 - \frac{1}{2} = 6(\frac{1}{2} + 1) + B(2*\frac{1}{2} - 1) \] On simplifying, we get B = -5. Therefore, the constants A and B are 6 and -5 respectively.

Step 4: Write the Final Solution

Substitute the values of A and B back into our partial fraction decomposition: \[ \frac{5-x}{2x^{2}+x-1} = \frac{6}{2x - 1} - \frac{5}{x + 1} \] This is the partial fraction decomposition of the given rational expression. It can be graphically verified through a graphing tool.

Key Concepts

Factoring Polynomials

One of the foundational skills necessary for algebra and calculus is the ability to factor polynomials. Factoring means breaking down a complex expression into simpler factors that, when multiplied together, return to the original expression. Let's take the expression from our exercise, \(2x^2 + x - 1\). To factor this, one needs to look for two numbers that multiply to give the product of the coefficient of \(x^2\) and the constant term (here, -2), and add up to the coefficient of \(x\) (here, 1).

In our case, those numbers are 2 and -1 because \(2 * -1 = -2\) and \(2 + -1 = 1\). This leads us to split the middle term and factor by grouping, eventually resulting in the expression being factored as \((2x - 1)(x + 1)\).

To practice factoring polynomials, students can start with the distributive property a(b + c) = ab + ac and reverse it to factor expressions. Puzzles, such as finding two numbers that fit the described criteria, can make this topic more engaging and manageable.

Rational Expressions

When dealing with fraction-like expressions where the numerator and the denominator are both polynomials, we are working with rational expressions. These expressions, such as \(\frac{5-x}{2x^2+x-1}\), require careful handling, particularly when simplifying, adding, subtracting, multiplying, or dividing.

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions, which are easier to integrate, differentiate, or sometimes even graph. After factoring the denominator, the process involves setting up a system of equations to solve for the unknown constants in the numerators of the simpler fractions. This technique is particularly helpful with integration in calculus.

Tips for students: always ensure the polynomial in the denominator is properly factored before starting the decomposition and work through the algebra systematically to solve for the unknown constants. We see this clearly in the step by step breakdown of the exercise where A and B are identified after setting up the equations.

Graphing Utility

Graphing utilities, such as online graphing calculators or software, can be incredibly effective in visualizing mathematical functions, including rational expressions. These tools not only provide graphical representation but can also help validate algebraic manipulations like the partial fraction decomposition.

For example, after obtaining the partial fraction decomposition \(\frac{6}{2x-1} - \frac{5}{x+1}\), we can input both the original rational expression \(\frac{5-x}{2x^2+x-1}\) and the decomposed form into a graphing utility. If done correctly, the graphs will coincide, proving the decomposition accurate.

Encouraging students to cross-verify their algebraic work with a graphing utility reinforces their understanding and provides a visual affirmation of the concepts at play. It demonstrates the tangible applications of abstract algebraic operations and brings the mathematical expressions 'to life' through their graphical plots.