91Ó°ÊÓ

To understand partial fraction decomposition, imagine you have a complex fraction—like breaking down a big problem into smaller parts. This is a method in algebra used to express a rational function as a sum of simpler fractions.
In our exercise, the goal is to decompose \(\frac{-3x+2}{(x-1)(x-\frac{1}{2})}\) so we can easily work with each individual part. We express this as:

• \(\frac{A}{x-1}\)
• \(\frac{B}{x-\frac{1}{2}}\)

This makes analysis straightforward, especially when it comes to integration or finding series expansions. By multiplying through by the denominator \((x-1)(x-\frac{1}{2})\), we eliminate the fractions temporarily to establish easy-to-solve equations for \(A\) and \(B\). In this example, solving these gives us \(A = 2\) and \(B = -4\). This step simplifies complex expressions, which is particularly useful in calculus.

The quadratic formula is a well-known method for finding roots of a quadratic equation \(ax^2 + bx + c = 0\). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this exercise, the quadratic formula is essential for factoring the denominator \(2x^2 - 3x + 1\). By inserting the coefficients \(a = 2\), \(b = -3\), and \(c = 1\), we find the roots \(x = 1\) and \(x = \frac{1}{2}\).
Understanding these roots helps us factor the quadratic expression into \((x-1)(x-\frac{1}{2})\), an essential step toward partial fraction decomposition.
This technique not only aids in algebra but also serves as a foundation for solving real-world problems involving parabolas and other quadratic scenarios.

Series expansion expresses a function as a sum of infinite terms. The Taylor series is a particular kind of series expansion centered around a certain point—in this case, 0. In our problem, we aim to find the Taylor series of each partial fraction obtained:

• For \(\frac{2}{x-1}\), it's rewritten as \(-2 \cdot \frac{1}{1-x}\), where we apply the geometric series formula \(\sum_{n=0}^{\infty} x^n\).
• For \(-\frac{4}{x-\frac{1}{2}}\), it's rewritten as \(-8 \cdot \frac{1}{1-2x}\) by using the geometric series formula \(\sum_{n=0}^{\infty} (2x)^n\).

Combining these series expansions, you arrive at:\[-2 \cdot \sum_{n=0}^{\infty} x^n - 8 \cdot \sum_{n=0}^{\infty} (2x)^n\]This step is fundamental in approximating functions and understanding their behavior near certain points.

A rational function is described as the quotient of two polynomials. Generally expressed as \(\frac{P(x)}{Q(x)}\), these functions are characterized by potential asymptotes or holes in their graph where the denominator (\(Q(x)\)) is zero.
In the exercise, the rational function given is \(\frac{-3x+2}{2x^2 - 3x + 1}\). To simplify this, the function is decomposed into partial fractions—a typical approach for handling complex rational expressions.
Rational functions appear in various fields of engineering and physics, often representing real-life situations where ratios are analyzed, such as speed (distance/time). Understanding this concept is crucial for both algebraic manipulations and interpreting graphs.