Factorization, sometimes also known as factoring consists of writing a number or another mathematical object as a product of several factors. Usually, factors are smaller or simpler objects of the same kind. It is an important concept in algebra. Here we will see some factorization formula. Let us learn it!
Concept of Factorization
In Mathematics, factorization is defined as the breaking or decomposition of an entity into a product of another entity, or factors. After the multiplication of factors together it will give the original number.
It is the resolution of an integer or polynomial into their factors. In the factorization method, we reduce the algebraic or quadratic equation into its simpler form. The factors of any equation can be an integer, a variable or algebraic expression itself.
Also, numbers can be factorized into different combinations. There are different ways to do factorization. Finding the factors of an integer is an easy method but finding the factors of algebraic equations is comparatively tough.
Factorization is not considered meaningful within number systems possessing division normally. For example, real or complex numbers. Since anyone can be trivially written as whenever it is not zero. Thus a meaningful factorization for a rational number or function can be obtained by writing it in its lowest terms both numerator and denominator.
Factorization in Algebra
The numbers 1, 2, 3, 4, 6, and 12 are all factors of 12 because they can be divided by 12 completely. It is an important process in algebra and is called Algebra factorization.
Some Factorization Formula:
In algebra for doing factorization various methods are available. Sometimes algebraic identities help a lot for solving such problems. Some of them are as follows:
- \(( m+p )^2 = m^2 + p^2 + 2mp\)
- \(( m-p )^2 = m^2 + p^2 – 2mp\)
- \(( m^2 – p^2 ) = (m+p) (m-p)\)
- \(( m+p )^3 = m^3 + p^3 +3mp (m+p)\)
- \(( m-p )^3 = m^3 – 3m^2p+3p^2m – p^3\)
- \(( a+b+c )^2 = a^2 + b^2 + c^2+ 2ab + 2bc + 2ca\)
- \(( a-b-c )^2 = a^2 + b^2 + c^2 -2ab – 2ca + 2bc\)
Factorizing formulas in algebra is especially important when solving quadratic equations. Also, while reducing formulas we normally have to remove all the brackets. In particular cases, for example with fractional formulas, sometimes we can use factorization to shorten a formula. The term is something that is to be added or subtracted. The factor is something that is to be multiplied.
Here are some maths factorizations example questions and how to factorize the quadratic equations are explained in detail.
Q.1: Factorize the following equation, which is a quadratic type: \(x^2 + 7x + 6\)
Solution: The constant term is 6, which can be further written as the product of 1 and 6.
Here (1 + 6) is giving 7 which is midterm coefficient.
\(X^2 + 7x + 6 = (x+1) (x+6)\)
Remember that the order doesn’t matter in multiplication.
Therefore, its answer is (x + 6)(x + 1).
Q.2: Factorize the following equation, which is a quadratic type: \(3x + 6xy + 30 + 60y\)
Solution: It is simple problem of factorization.
3X + 6XY + 30 + 60Y
Taking suitable common terms,
3X(1+2Y)+ 30 (1+2Y)