Let us talk about a key mathematical word called "factor" before we get started with factorization. Factors are integers, algebraic variables, or algebraic expressions that perfectly divide the number or expression without leaving any remainder or residual. For Example:
1. The factors of 44= 2 × 2 × 11, the numbers 2, 4, and 11 are the factors of 44 and divide it without resulting in the formation of any remainder.
2. The factors of an algebraic expression 22xy = 2 × 11 × x × y.
When we divide an integer or a polynomial into its constituent parts or factors or other equations, which when multiplied together again yields the same polynomial or number is known as factorization.
For instance – The factors of 24 are 1 x 2 x 2 x 2 x 3; when we multiply them in any order, we again get the value as 24. A number or an algebraic statement is factorized when it is expressed or decomposed as a product of its prime factors or irreducible factors. We will learn how to factor an algebraic equation as a product of its components in this post. The components may be easily found by reading aloud previous instances of algebraic expressions 22xy. Consider complicated algebraic equations such as 2x 2 - 20xy + 10 = 34, x 2 – 2y = 0, etc. Determining their components is difficult. We'll need to use certain factorization techniques to discover factors for these phrases. They are listed as:
Example: 12xy + 4xz Dividing each term 12xy = 2 × 2 × 3 × x × y 4xz = 2 × 2 × x × z The common factors in these terms are 2 × 2 × x Therefore, 12xy + 4yz = (2 × 2 × 3 × x × y) + (2 × 2 × x × z) = 4x (3y + z)
There are instances when all the terms of an expression have no common component. However, these terms may be regrouped so that each group has a common component. When we do this, a common factor emerges from all of the groups, resulting in the factorization of the expression that is necessary.
Using Identities:
By placing algebraic expressions in the form of appropriate identities, a number of algebraic expressions can be factored. These identities are:
1. m 2 + 2mn + m 2 = (m + n) 2
2. m 2 – 2mn + n 2 = (m – n) 2
3. m 2 – n 2 = (m + n) (m – n)
Whenever we find an equation relating to these identities, we can directly apply them to find the factors of the problem given to us.
Whenever we encounter the algebraic expressions in the form of mx 2 + px + q, we should always remember that since it is a quadratic equation, it has two roots or factors ‘k’ and ‘l’, such that the addition of both the roots is equal to the coefficient of x, i.e., ‘p’ and the product of the roots is equal to the constant term ‘q’.
These two properties are used to divide any given equation to the form – mx 2 + (k + l) x + kl = 0. This can easily be solved by resolving the bracket into two terms and then taking common, which gives the values (x + k) and (x + l).