Partial Fractions: Simplifying 2nd Set to 1st

[Rach123]

I don't fully understand the logic of this example:

For, 4x^2-3x+5/(x-1)^2(x+2) we need: A/(x-1)^2+B/(x-1)+C/(x+2)
It is also correct to write Ax+B/(x-1)^2 + C/(x+2) but the fractions are not then reduced to the simplest form.

How do the 2nd fractions simplify to give the 1st set of fractions?

 

[dynamicsolo]

I don't fully understand the logic of this example:

For, 4x^2-3x+5/(x-1)^2(x+2) we need: A/(x-1)^2+B/(x-1)+C/(x+2)
It is also correct to write Ax+B/(x-1)^2 + C/(x+2) but the fractions are not then reduced to the simplest form.

How do the 2nd fractions simplify to give the 1st set of fractions?

They don't exactly: the constants in the second set will be a little different. They are just saying that the two sums are equivalent, but the first sum is preferred for certain purposes. (I don't agree that the first set is simpler.)

What you would actually find for the first two terms in the first set would be

[A/(x-1)^2] + [B/(x-1)] = [A/(x-1)^2] + [{B(x-1)}/(x-1)^2] =

[{A+Bx-B}/(x-1)^2] ;

they then consolidated the A-B in the numerator into a single constant and relabeled the coefficients. But the A and B in the second set will not be then same as they are in the first set (which is a usage I find a bit sloppy)...

You are correct in saying that you can't just rearrange the second set to get the first one.

 

[Architeuthis Dux]

In order to understand the logic behind simplifying from the 2nd set of fractions to the 1st set, it is important to first understand the concept of partial fractions. Partial fractions is a method used to simplify and integrate rational functions, which are functions that can be expressed as a ratio of two polynomials. The goal of partial fractions is to break down a complex rational function into simpler, more manageable parts.

In the given example, we have a rational function with a denominator of (x-1)^2(x+2). In order to simplify this function, we need to express it as a sum of simpler fractions, each with a distinct denominator. This is where the 2nd set of fractions comes in - it is a decomposition of the original function into simpler fractions.

The process of finding the 2nd set of fractions involves solving for the unknown constants A, B, and C using algebraic methods. Once the constants have been determined, we can then rewrite the 2nd set of fractions as a single rational function. This is what is represented in the first equation given in the example.

However, as mentioned in the second equation, the fractions are not reduced to the simplest form. In order to simplify further, we need to factor out the common denominator of (x-1)^2(x+2) and combine like terms. This results in the 1st set of fractions, which is a simplified version of the original rational function.

To summarize, the 2nd set of fractions represents the decomposition of a complex rational function into simpler fractions. By solving for the unknown constants and simplifying, we arrive at the 1st set of fractions, which is a simpler form of the original function. This process is an important tool in simplifying and integrating rational functions in mathematics and science.