It is always better to start from scratch and know how the process in-built together and incorporated at places than to start working and return back to the basics.
What does the term Factorization mean?
It is the process of finding out factors of a polynomial or any equation. It is moreover like splitting of a complex term into two other numbers whose multiplication product if equal to the main term.
It can also be defined as the decomposition of a mathematical equation and is not necessarily applicable to polynomials, but also for Matrices and numbers
Why should one Factorize?
The main objective of factorizing is to reduce the complexity of the problem. The end results of Factorizing are something which cannot be further divided, thus breaking the equation to its simpler form.
The factorizing terms in their form can further be multiplied again to retain the original equations. This is also known as the expansion of polynomials. It is the best way to verify the solutions obtained after the factorizing process.
When should we Factorize?
Before moving on to factorization let us have a look at the factor theorem:
The factor theorem states that when a polynomial f(x) has a factor that is (x-k), it is only when f(x)=0
This means that the value of k that is obtained is a root of the polynomial.
Other uses of the Factor theorem are:
The factor theorem is used to remove known zeros from a polynomial while leaving all unknown zeros fixed, thus producing a lower degree polynomial whose zeros are easier to solve.
The method is:
1. Use the factor theorem to conclude that the term (x-a) is a factor of f(x).
2. Solve the polynomial g(x) =f(x)/f(x-a), for example using Long Division method of Polynomials or Synthetic Division method.
3. Say that any root x is not equal to an of f(x) =0 is a root of g(x)=0.
4. Since the polynomial degree of g is one less than that of the degree of f, it is "simpler" to find the remaining zeros.
Few Common Methods of Factorization
Listed below are the few commonly used methods for factorization:
Let us have a quick look at them
1. Highest Common Factor
The highest common factor of algebraic expressions is useful in factorization; and it is the product of the common prime factors, which includes both common numerical and algebraic factors.
2. Using the Common factor
3. Using the Difference of the square formula
By using the above method, you can easily find the factors of a number.
Real Life applications of Factorization
Commonly used in finance and accounting where the present value of the assets are determined and for stock evaluation. Its uses are also traced in bond trading and calculations.
For complex and high ordered equations in derivations and formulae method of factorization is needed to reduce the complexity of the problem, thus getting a valid solution or a better and simpler form of the equation. Also, high tech algorithms use the basics of Factorizing.
Integration sums and wave equations always end up unexpectedly in a quadratic form and then the only factorization can reach for your help. It is also widely used for solving numerical and calculation unknown values when in higher than or equal to 2nd order format.