Problem 22 - Precalculus Mathematics 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, factor the denominator of the rational function. The denominator is \(2x^2 - x\). This can be factored as \(x(2x - 1)\).

Step 2: Set Up Partial Fraction Decomposition

Express \(\frac{8x-3}{x(2x-1)}\) as a sum of two simpler fractions. Since the denominator factors as \(x(2x - 1)\), we can write it as \(\frac{A}{x} + \frac{B}{2x-1}\), where \(A\) and \(B\) are constants to be determined.

Step 3: Clear the Denominator

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

Step 4: Expand and Match Coefficients

Expand the right side: \[8x - 3 = 2Ax - A + Bx.\]Combine like terms: \[(2A + B)x - A.\]Compare coefficients with \(8x - 3\) gives the system of equations: \[2A + B = 8,\]\[-A = -3.\]

Step 5: Solve for Constants

Solve the system of equations to find \(A\) and \(B\). From \(-A = -3\), we get \(A = 3\). Substitute \(A = 3\) into \(2A + B = 8\):\[2(3) + B = 8\]\[6 + B = 8\]\[B = 2.\]

Step 6: Write the Partial Fraction Decomposition

Substitute \(A\) and \(B\) into the partial fractions:\[\frac{8x - 3}{x(2x - 1)} = \frac{3}{x} + \frac{2}{2x - 1}.\]

Key Concepts

Rational Function

A rational function is a type of mathematical expression that is formed by the division of two polynomials. In simpler terms, it is a fraction where both the numerator and the denominator are polynomials. For instance, in the expression \(\frac{8x-3}{2x^2-x}\), \(8x-3\) is the numerator, and \(2x^2-x\) is the denominator.

Rational functions are crucial in calculus and algebra because they can be used to model real-world situations involving ratios and proportions. Understanding how to manipulate and simplify these functions is essential. One core skill related to rational functions is transforming them into more manageable expressions through techniques like partial fraction decomposition. This process helps to simplify the function for easier integration or solving.

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler "factors" that, when multiplied together, give back the original polynomial. It is similar to finding the prime numbers that multiply to a given number.

In the given exercise, the polynomial in the denominator \(2x^2 - x\) is factored into simpler terms. The expression \(2x^2 - x\) can be factorized as \(x(2x - 1)\). Here's how it works:

• Identify the greatest common factor (GCF) that can be divided out from all terms in the polynomial.
• In \(2x^2 - x\), both terms have \(x\) as a common factor.
• Factor out \(x\) from each term, resulting in \(x(2x - 1)\).

Factoring is an invaluable technique in algebra for simplifying expressions and solving equations.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. Solving a system means finding values for the variables that satisfy all the equations simultaneously.

In the partial fraction decomposition, after clearing the denominator, we derived the system of equations:

• \(2A + B = 8\)
• \(-A = -3\)

The solution to this system provides the values for constants \(A\) and \(B\), which are necessary to complete the decomposition. Solving the system involves:

• First, deducing from \(-A = -3\) that \(A = 3\).
• Substituting \(A = 3\) into \(2A + B = 8\) to find \(B = 2\).

Understanding how to set up and solve these equations is important when working with algebraic fractions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators such as addition or multiplication. They are foundational in algebra and can represent quantities in a variety of ways.

In the process of finding partial fractions, we dealt with algebraic expressions like \(8x - 3\) and \(2x^2 - x\). To manipulate and simplify these expressions, we followed these steps:

• Express the complex fraction \(\frac{8x-3}{2x^2-x}\) as a sum of simpler fractions.
• Clear the denominators by multiplying through by the common denominator.
• Rearrange and combine like terms to solve for unknowns.
• Substitute back to express the original fraction as a sum of its parts, \(\frac{3}{x} + \frac{2}{2x-1}\).

Mastering these operations on algebraic expressions is key to solving equations and understanding algebraic functions.