Problem 63 - 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. $$\frac{5-x}{2 x^{2}+x-1}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{5-x}{2x^2+x-1}\) is \(\frac{1}{2x-1} - \frac{2}{x+1}\). The correctness of these fractions has been checked algebraically and graphically.

Step 1: Factorization

The first step is to decompose the denominator so it can be expressed as two linear factors. In its current form it is \(2x^2+x-1 = 0\). Using factorization we find it can be factored as \(2x^2+x-1 = (2x-1)(x+1)\).

Step 2: Formation of Partial Fractions

The next step is to express the given fraction as the sum of two simpler fractions. The decomposed fraction is of the form \(\frac{A}{(2x-1)}+ \frac{B}{(x+1)}\). To get the values of the A and B multiplicative factors, equate this to the given fraction and solve for A and B.

Step 3: Calculating the Values of A and B

Performing the above, we get \(\frac{5-x}{2x^2+x-1} = \frac{A}{(2x-1)}+ \frac{B}{(x+1)}\). Multiplying through by \((2x^2+x-1)\), it gives \(5-x = A(x+1) + B(2x-1)\). By equating coefficients of like terms, we get two equations and on solving them, we get A = 1 and B = -2.

Step 4: Substitution and Formation of Partial Fraction Decomposition

Substituting A and B into the partial fractions, we get \(\frac{5-x}{2x^2+x-1} = \frac{1}{2x-1} - \frac{2}{x+1}\). This is the partial fraction decomposition for the given rational expression.

Step 5: Verification

The verification of the above result is done using a graphing utility to plot the initial expression and the obtained partial fractions in the same graph, by which we can observe the matching graphs.

Key Concepts

Rational Expression

A rational expression is simply a fraction where both the numerator and the denominator are polynomials.
These expressions are fundamental in algebra and calculus, as they help represent relationships between quantities.
Understanding rational expressions is essential for simplifying complex algebraic fractions and solving equations that involve them. Some key aspects of rational expressions include:

• Simplification: Simplifying rational expressions involves reducing them to their simplest form by factoring the numerator and the denominator and canceling out common factors.
• Domain: The domain of a rational expression includes all real numbers except those that make the denominator zero.
• Operations: You can add, subtract, multiply, and divide rational expressions using similar techniques as with numerical fractions.

Recognizing patterns within the polynomials will assist in correctly identifying and simplifying expressions more efficiently.

Factorization

Factorization is the process of breaking down a polynomial into simpler factors that, when multiplied together, give the original polynomial.
In the context of partial fraction decomposition, factorization helps simplify the denominator to analyze the rational expression.
This is a crucial first step before decomposing into partial fractions.Key points to remember about factorization include:

• Prime Factors: Polynomials are expressed in terms of irreducible factors over a specific field (like the set of real numbers).
• Methods: Some common factorization methods include dividing polynomials, using the quadratic formula, or applying synthetic division.
• Applications: Factorization is not only used in partial fraction decomposition but is also helpful in solving equations and simplifying expressions.

In our exercise, factorizing the denominator \(2x^2 + x - 1\) to \((2x-1)(x+1)\) allowed us to proceed to the next steps wherein we explore partial fraction decomposition.

Algebraic Verification

Algebraic verification involves checking the correctness of the partial fraction decomposition.
This step ensures that the decomposition accurately represents the initial rational expression.
You achieve this by equating the original expression to the sum of partial fractions and confirming the equality through algebraic manipulations.The process involves:

• Reconstruction: Express the original rational expression as a sum of the partial fractions found after decomposition.
• Equating: Multiply through by the original denominator to eliminate fractions, producing a polynomial identity.
• Comparison: Compare coefficients on both sides of the identity to solve for the unknown values (like \(A\) and \(B\) in our practice step).

Once you've confirmed that both sides of the equation match, you've successfully verified the partial fraction decomposition.

Graphing Utility

Using a graphing utility is a valuable technique for visual verification.It helps you visually verify the results by plotting both the original rational expression and its partial fraction components.In our example:

• Graph the original function \(\frac{5-x}{2x^2+x-1}\).
• Graph each partial fraction function, such as \(\frac{1}{2x-1} - \frac{2}{x+1}\).
• Compare: In the same graphing window, look for overlapping curves, indicating that both representations are equivalent.

This approach not only confirms the algebraic work but also enhances understanding through different representations.