Division of Algebraic Expressions
Division of a monomial by another monomial
Division of a monomial by another monomial:
i) Division of 9x by 3: 2
2
3(3x )
2 2
9x ÷3 = = 3x
3
ii) Division of 6x 2
y by 2y:
2 2
6x y 2y(3x )
2 2
6x y ÷ 2y = = = 3x
2y 2y
Division of a polynomial by a monomial
A polynomial 2x 3
+ 4x
2
+ 6x is divided by monomial 2x as shown below:
3 2
(2x +4x +6x) 3 2
2x 4x 6x
2
= + + = x + 2x + 3
2x 2x 2x 2x
Division of a polynomial by a polynomial
Long division method is used to divide a polynomial by a polynomial.
Example:Division of 3x + 3x − 5 by (x − 1) is shown below:
2
Introduction to Factorisation
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 is 2 × 3 × 7, where 2, 3 and 7 are factors of 42.
Algebraic expressions
An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables
(like x or y) and operators (like add,subtract,multiply and divide). For example: x + 1, p - q, 3x,
2x+3y, 5a/6b etc.
Factors of algebraic expressions and factorisation
An irreducible factor is a factor which cannot be expressed further as a product of factors.
Algebraic expressions can be expressed in irreducible form.
Example: 7 × (x + 3) is written as 7 × x × (x + 3), where 7, x and (x + 3) are the irreducible
factors of expression 7x(x + 3).
Method of Common Factors
Factorisation by common factors
To factorise an algebraic expression, the highest common factors are determined.
Example: Algebraic expression −2y + 8y can be written as 2y(−y + 4), where 2y is the highest
2
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 are grouped
together.
Example:
12a + n − na − 12
= 12a − 12 + n − na
= 12(a − 1) − n(a − 1)
= (12 − n)(a − 1)
(12 − n) and (a − 1) are factors of the expression 12a + n − na − 12
Method of Identities
Algebraic identities
The algebraic equations which are true for all values of variables in them are called algebraic
identities.
Some of the identities are,
2 2 2
(a + b) = a + 2ab + b
2 2 2
(a − b) = a − 2ab + b
2 2
(a + b)(a − b) = a −b
Factorisation using algebraic identities
Algebraic identities can be used for factorisation
Example:
(i) 9x2
+ 12xy + 4y
2
2 2
= (3x) + 2 × 3x × 2y + (2y)
2
= (3x + 4y)
(ii) 4a 2
−b
2
= (2a − b)(2a + b)
Visualisation of factorisation
The algebraic expression x 2
+ 8x + 16 can be written as (x + 4) . This can be visualised as
2
shown below:
