Problem 20 - Precalculus: Mathematical for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{8 x-3}{2 x^{2}-x}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{3}{x} + \frac{2}{2x - 1}\).

Step 1: Factor the Denominator

First, identify the denominator of the rational function: \(2x^2 - x\). This can be factored by pulling out a common factor of \(x\), resulting in \(x(2x - 1)\). So, the rational function becomes \(\frac{8x - 3}{x(2x - 1)}\).

Step 2: Set Up the Partial Fractions

The fraction is now ready to be decomposed into partial fractions. Since the denominator consists of distinct linear factors, write it as: \(\frac{A}{x} + \frac{B}{2x - 1}\).

Step 3: Clear the Denominator

Multiply both sides by the common denominator \(x(2x - 1)\) to eliminate the denominators: \[8x - 3 = A(2x - 1) + Bx\].

Step 4: Expand and Collect Like Terms

Expand the right side of the equation: \[8x - 3 = 2Ax - A + Bx\].Combine like terms to get a single expression on the right side: \[8x - 3 = (2A + B)x - A\].

Step 5: Solve for Coefficients

To find \(A\) and \(B\), equate the coefficients from both sides. First, equate the coefficients of \(x\): \(2A + B = 8\).Next, set the constant terms equal to each other: \(-A = -3\) which means \(A = 3\).Substitute \(A = 3\) into the first equation: \(2(3) + B = 8\) which simplifies to \(6 + B = 8\), hence \(B = 2\).

Step 6: Write the Final Decomposition

Substitute the values of \(A\) and \(B\) back into the partial fractions: \[\frac{8x - 3}{x(2x - 1)} = \frac{3}{x} + \frac{2}{2x - 1}\].This is the partial fraction decomposition of the original rational function.

Key Concepts

Rational Function

A rational function is essentially a quotient of two polynomials. Imagine it like a fraction where both the numerator and the denominator are polynomials. These functions can take on complex behaviors, especially as the degree of the polynomials increases. The degree of a polynomial is the highest exponent of the variable in that polynomial.
When dealing with rational functions, it's important to understand that they can be expressed in simpler parts through a process called partial fraction decomposition. This is especially useful for integrating or simplifying expressions. By breaking down the function into simpler fractions, the operations on these functions become more manageable.
In the given exercise, \[\frac{8x-3}{2x^2-x}\] is a rational function where the numerator is a first-degree polynomial and the denominator is a second-degree polynomial.

Factoring Polynomials

Factoring polynomials is a technique used to simplify or decompose polynomial expressions. When you factor a polynomial, you're writing it as a product of simpler, lower-degree polynomials. This is crucial in many areas of mathematics, especially when dealing with rational functions and partial fractions.
In our problem, we needed to factor the denominator of the rational function \[2x^2 - x\]. Factoring involves looking for common factors or making the expression a product of its roots. Here, the expression is factored by taking out the common factor of \(x\), resulting in \[x(2x - 1)\]. This step allows us to break the function into partial fractions later.
Factoring simplifies complex expressions and is a necessary step in transforming rational functions into partial fractions.

Linear Factors

Linear factors are polynomial expressions of the first degree, meaning they can be written in the form \(ax + b\). These factors play an essential role in partial fraction decomposition since they determine how a rational function can be expressed as a sum of simpler terms.
In our exercise, the factored denominator \[x(2x - 1)\] is composed of linear factors. The partial fraction decomposition relies on expressing the original rational function as a sum of terms like \(\frac{A}{x}\) and \(\frac{B}{2x - 1}\). Each term corresponds to one linear factor of the denominator.
Understanding and identifying linear factors make it easier to set up partial fraction equations and solve for the unknown coefficients, simplifying the originally complex rational function.

Equating Coefficients

Equating coefficients is a method used to find unknown constants in mathematical expressions. It's particularly useful in partial fraction decomposition to solve the values of coefficients in simplified fractions.
In step 5 of the solution, the expanded expression \[(2A + B)x - A\] is set equal to \[8x - 3\]. By equating the coefficients of like terms (coefficients of \(x\) and the constant terms), we can solve for the unknowns \(A\) and \(B\).

• First, compare the coefficients of \(x\): here, \[2A + B = 8\].
• Next, compare the constant terms: \[-A = -3\], which resolves to \(A = 3\).
• Substitute \(A = 3\) into the equation for \(B\): \[6 + B = 8\], giving \(B = 2\).

This ensures that both sides of the equation match perfectly for all values of \(x\), confirming our partial fraction decomposition is correct.