Algebra Calculators

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Programmers and Teachers: Nikkolas
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# Partial Fraction Decomposition Calculator

## Lesson on Partial Fraction Decomposition

### Why Decompose a Fraction?

Sometimes when working with a fraction, whether it be an application in engineering, science, or pure mathematics, we must “expand” it out fully to the non-simplified version of itself. This is often because the fraction has a denominator that is complex and prevents us from finding whatever solution we are after. Complex denominators can make adding, subtracting, multiplying, and dividing fractions very difficult.

Partial fraction decomposition can be thought of as the opposite of simplifying a fraction. Note that “simplifying” is used here in its classical algebra definition. Performing partial fraction decomposition can make problems simpler to solve, even though the fractions have become expanded. Algebraically, the fraction may be less simplified, but the entire expression may be more open to simplification afterwards.

### Examples

Let’s take a look at a couple examples of fractions with partial fraction decomposition applied to them.

Example 1:
Fraction = (3x + 11)/[(x – 3)(x + 2)]
Partial fraction decomposition = [A/(x – 3)] + [B/(x + 2)]

Example 2:
Fraction = 1/[(x – 1)(x + 1)]:
Partial fraction decomposition = [A/(x – 1)] + [B/(x + 1)]

## How the Calculator Works

The calculator above uses a computer algebra system to symbolically calculate the partial fraction composition. Because the calculation is symbolic, it preserves exact values rather than making approximations like a numerical method/routine would. It does the symbolic calculations very quickly.

Behind the scenes, there is very complex and lengthy code that powers the computer algebra system. However, the code can be broken down into very many simple logic statements. It follows a similar process to what humans use when hand calculating partial fraction decomposition. It uses the same algebra rules that we follow in school and our professions, with the advantage of having a very fast computer processor!

Result :

Worksheet 1