Simplification

Fractions and stuff

Greatest common divisor

A common divisor is any number c, such that c|a \land c|b. If M is the set of all common divisor, we would define the gcd as m = max\{M\}.

Euclidean Algorithm

To compute the gcd:

1. Convert all inputs to absolute values
2. Calculate larger mod smaller
3. Take the smaller number and the calculated rest as new inputs
4. Repeat until the rest is 0
5. At that point the smallest number is the gcd

Coprime

German: Teilerfremd

a and b are co-prime if their gcd is 1. If a and b are co-prime the fraction \frac{a}{b} cant be reduced.

I'd factor this in

Factoring is the process of breaking down a mathematical expression (usually a polynomial) into a product of simpler expressions called factors.

Greatest common factor

A factor are numbers we can multiply to another number. A common factor that can be found in all terms of an expression. The greatest common factor is the largest of these factors.

To factor out the GCF:

1. Find the GFC of all coefficients
2. Find the lowest power of each variable
3. Factor out the GCF

Difference of Squares

The difference of square formula is an algebraic form of the equation used to express the differences between two square values.

This is expressed as: a^2 - b^2

Factoring the difference gives us: a^2 - b^2 = a^2 - ab + ab - b^2 = (a + b) (a - b)

To factor out the difference of squares:

1. Determine if the terms have a GCF and factor it out.
2. Apply the formula (a^2 - b^2 = a^2 - ab + ab - b^2 = (a + b) (a - b)) to the numbers that will produce the same results.

Perfect Square Trinomial

A perfect square trinomial is the result of squaring a binomial expression.

Example: a^2 + 2ab + b^2

To factor out a perfect trinomial:

1. Check if the middle term is positive or negative.
2. If the term is positive then the factors are (a+b)(a+b), otherwise the factors are (a-b)(a-b) or (b-a)^2.

Trinomial Factoring

TODO!

Validations Method

TODO!