What is the difference between identity and equation?

An identity is an expresion that is true for any value of the letters:

x + 1 = 2x + 2              (is true for any value of x)
(a+b)·(a+b) = a2 + 2ab + b2            (is true for any value of a and b)

An equation is valid only for a small number of values of the letters:

x + 3 = 5                    (is true only for x = 2)
x2 - 1 = 0                  (is true only for x=1 and x=-1)

Factorization techniques for algebraic expresions

Main factorization techniques for polinomials or algebraic expresions:

- First of all, you can look for a common factor in all summands. If it exists, you can pull it out with the distributive property.

- You can use identities like the square of an addition, the square of a subtraction, and addition by subtraction.

- You can factorize a polinomial using ruffini rule. By this method, you will find the solutions of that equation, and this will be easy to make it in a multiplication way.

- If the equation is in the second degree, it is not neccesary to use Ruffini rule, because we can find the solutions using the main formula for that kind of equations.

What does "factorize" mean?

When we learn maths, we have receive the problem to factorize some expresions or numbers, but what does it means?

Factorize is the process by which we convert anything (a number or algebraic expresion) in a multiplication.

So, if we must factorize a number, we will transform that number in a multiplication of primes numbers:

factorization of 24 = 2·3·4

And factorization of an algebraic expresion, is another expresion transformed in a multiplication instead of an addition:

x2 + 2x + 1 = (x+1)·(x+1)

Remember: factor is a part of a multiplication, in the same way that a summand is a part of an addition.

How to know if a fraction is irreducible?

The best way to know is a fraction is irreducible is:

- Factorize both numerator and denominator.

- If there is no factor equal in numerator and denominator, the fraction is irreducible.

- If there is any factor equal in both numerator and denominator, just erase the same factor in both parts and you will find de irreducible fraction.

Another way to say this is that "numerator and denominator are mutual primes" (they don´t have any factor in common).

How to find quickly an irreducible fraction?

If you have a fraction to simplify, it is normal to try some numbers dividing for numerator and denominator in order to find the irreducible fraction.
The best and quick way to proceed to avoid loss of time is:

- Calculate the greatest common divisor between numerator and denominator.

- Now divide numerator and denominator by this number. The fraction you have obtain is the irreducible one.