Problem 2 - Thomas Calculus

Question

Expand the quotients in Exercises \(1-8\) by partial fractions. $$ \frac{5 x-7}{x^{2}-3 x+2} $$

Step-by-Step Solution

Answer

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

Step 1: Factor the Denominator

The first step is to factor the quadratic expression in the denominator. Given the denominator is \(x^2 - 3x + 2\), we can factor it into \((x - 1)(x - 2)\).

Step 2: Set Up Partial Fraction Decomposition

Write the fraction as a sum of partial fractions. For the expression \(\frac{5x-7}{(x-1)(x-2)}\), we can decompose it as \(\frac{A}{x-1} + \frac{B}{x-2}\).

Step 3: Clear the Denominator

Multiply both sides of the equation by the factored denominator to eliminate the fractions: \[(5x - 7) = A(x - 2) + B(x - 1)\]

Step 4: Solve for Coefficients A and B

Expand the right side and collect like terms:\[A(x - 2) + B(x - 1) = Ax - 2A + Bx - B = (A + B)x - 2A - B\]Now, equate the coefficients with those on the left side:\[A + B = 5\]\[-2A - B = -7\]

Step 5: Solve the System of Equations

Solve the equations from Step 4:- From \(A + B = 5\), we can express \(B = 5 - A\).- Substitute \(B = 5 - A\) in the second equation: \[-2A - (5 - A) = -7\] Simplify to \(-2A - 5 + A = -7\) leading to \( -A - 5 = -7\) so \(-A = -2\), giving \(A = 2\).- Substitute \(A = 2\) back into \(B = 5 - A\) to find \(B = 3\).

Step 6: Write the Final Partial Fraction Expansion

With \(A = 2\) and \(B = 3\), substitute back into the partial fraction form:\[\frac{5x-7}{(x-1)(x-2)} = \frac{2}{x-1} + \frac{3}{x-2}\]

Key Concepts

Quadratic Factorization

Quadratic factorization is a crucial technique in algebra, especially when dealing with expressions that involve quadratic terms. A quadratic expression typically takes the form \(ax^2 + bx + c\). The goal of factorizing such an expression is to express it as a product of simpler linear factors, whenever possible.

To factorize \(x^2 - 3x + 2\), we look for two numbers that multiply to the constant term (2) and add up to the coefficient of the linear term (-3). These numbers are -1 and -2. Therefore, the expression \(x^2 - 3x + 2\) can be written as \((x - 1)(x - 2)\). This process makes it easier when performing partial fraction decomposition as it simplifies the denominator into linear factors.

Here are a few tips for successful quadratic factorization:

• Identify the standard form of your quadratic expression.
• Find two numbers that multiply to the product of the leading coefficient and constant term, and sum to the middle term.
• Use these numbers to break down the middle term and factor by grouping.

The ability to factor quadratic expressions quickly helps simplify many algebraic processes, such as solving equations and performing partial fraction decomposition.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In the context of partial fraction decomposition, a system of equations arises when solving for the unknown coefficients of the decomposed fractions.

In our example, after decomposing \(\frac{5x-7}{(x-1)(x-2)}\) into partial fractions \(\frac{A}{x-1} + \frac{B}{x-2}\), we clear the denominators and form a system to determine values of \(A\) and \(B\). The system that emerges is:

• \(A + B = 5\)
• \(-2A - B = -7\)

To solve this system, we use substitution or elimination. Substitution involves expressing one variable in terms of another and then solving the resulting equations. For example, from the first equation, we find \(B = 5 - A\), and substitute this into the second equation to find \(A=2\), and consequently, \(B=3\).

Understanding how to solve systems of equations is essential in mathematics as it appears in various areas, such as when dealing with linear algebra and calculus.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Dealing with rational expressions is a common task in algebra and calculus.

Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions. This method is particularly useful for integration problems in calculus or when simplifying expressions.

In our exercise, the expression \(\frac{5x - 7}{x^{2} - 3x + 2}\) is decomposed into simpler fractions \(\frac{2}{x-1} + \frac{3}{x-2}\). These simpler pieces are easier to work with, especially when integrating them. The process involves:

• Factoring the denominator into simpler linear factors.
• Setting up the problem as a sum of fractions whose denominators are these simpler factors.
• Finding the coefficients of these fractions such that the original expression is satisfied when combined.

By using partial fraction decomposition, you can simplify the work involved in handling rational expressions, making mathematical operations more straightforward.