What are Factors?
An expression can be factorised into the product of its factors. These factors can be algebraic expressions, variables and numbers also.
Division of a monomial by another monomial
Division of a monomial by another monomial:
i) Division of 9x2 by 3:
9x2 ÷ 3 = 3(3x2) / 3 = 3x2
ii) Division of 6x2y by 2y:
6x2y÷2y = (6x2)y / 2y = 2y(3x2) / 2y = 3x2
Division of a polynomial by a monomial
A polynomial 2x3 + 4x2 + 6xis divided by monomial2xas shown below:
(2x3+4x2+6x)2x = 2x3 / 2x + 4x22x + 6x / 2x = x2+ 2x + 3
Division of a polynomial by a polynomial
Long division method is used to divide a polynomial by a polynomial.
Example:Division of 3x2 + 3x – 5 by(x – 1) is shown below:
Factors of natural numbers
Every number can be expressed in the form of product of prime factors. This is called prime factor form.
Example: Prime factor form of 42 is2 x 3 x 7, where 2, 3 and 7 are factors of 42.
An algebraic expression is defined as the mathematical expression which consists of variables, numbers, and operations. The values of this expression are the not constant.For example:x + 1, p – q, 3x, 2x+3y, 5a/6b etc.
Factors of algebraic expressions and factorisation
Anirreduciblefactor is a factor which cannot be expressed further as a product of factors.
Algebraic expressionscan be expressed in irreducible form.
Method of Common Factors
Factorisation by common factors
To factorise an algebraic expression, thehighest common factorsare determined.
Example: Algebraic expression -2y2 + 8y can be written as 2y(-y+4), where 2y is the highest common factor in the expression.
Factorisation by regrouping terms
In some algebraic expressions, it is not possible that every term has a common factor. Therefore, to factorise those algebraic expressions, terms having common factors aregroupedtogether.
= 12a + n – na – 12
(12-n) and (a-1)are factors of the expression12a+n-na-12
Method of Identities
The algebraicequations which are true for all values of variables in them are called algebraic identities.
Some of the identities are,
(a+b)2 = a2+ 2ab + b2
(a-b)2= a2– 2ab + b2
(a+b)(a-b)= a2 – b2
Factorisation using algebraic identities
Algebraic identities can be used for factorisation
Example: (i)9x2 + 12xy + 4y2
= ( 3x)2 + 2 x 3x x 2y+(2y)2
(ii)4a2 – b2 = (2a-b)(2a+b)
Visualisation of factorisation
The algebraic expression x2 + 8x + 16 can be written as (x+4)2. This can be visualised as shown below: