1. Introduction to Factorization


Introduction to Factorization

Type I-Factorisation by common factor

1. Consider the expression 3x3−15x2+12x.

In all the term, 3x is taken commonly outside.

(3x×x2)+(3x ×−5x)+(3x×4)


2.  Consider the expression 10xy+5x 3y.

Here the term 5xy is common in the expression. 

10xy+5x 3y


Type II-Factorisation by common binomial factor

Consider (n2+1)(m−n)+(m2+1)(m−n).

Take binomial factor from each term commonly outside.

(n2+1)(m−n) +(m2+1)(m−n)


=(m−n)(n 2+m2+2)

Type III-Factorisation by grouping

Consider the expression 3m2 +mn+3mn+n2.

Let us take the common factor for the first two terms separately and take the common factor second two terms separately. 




Type IV-Factorisation using identities

Use the following identities to factorise the expressions.


Type V- Factorisation of expression in Quadratic form

Procedure to factorise the expression

Step 1: Determine the coefficient a,b and c.

Step 2: Calculate the product of a and c. Product =ac and sum =b. Thus the middle coefficient is the sum and extreme product is the product value.

Step 3: Express the middle term as sum of two terms such that the sum satisfies the middle term and the product satisfies the extreme product.

Step 4:  Now group the expression into two factors by taking the common expression outside.

2. Division of Algebraic Expressions

Division of Algebraic Expressions

Introduction to division of algebraic expressions

There are four ways to divide an algebraic expression by another expression:

i) Dividing a monomial by monomial

ii) Dividing a polynomial by monomial

iii) Dividing a binomial by monomial

iv) Dividing a polynomial by polynomial

Division of an algebraic expression by a monomial

i) Division of a monomial by another monomial:

A monomial 40xy 2 is divided by another monomial 10y will result in 40xy2 /10y=4xy.

The result of dividing a monomial by another monomial will be a monomial.

ii) Division of a polynomial by a monomial:

Divide each term of the polynomial by the monomial to get the result of the division.

A polynomial −12xyz 3+60 is divided by a monomial 4z will result in: 3xyz 2+15z

Dividing any polynomial by a monomial will result in a polynomial.

The relation between the power of exponents and division of an algebraic expression by another algebraic expression:

The above law of exponents can be used to divide an algebraic expression by another, for instance: 4xy 2/2y=2xy 2−1  =2xy

Finding error

Let us look at the division of algebraic expression: (3x+2)/2=3x 

when the numerator is connected by the terms we cannot directly cancel the values from numerator and denominator.


Let us look for some general error that we have done while solving the exercises involves algebraic expressions.

1. 3(x+2)=3x+2 - In this case, we need to apply distributive law to find the result. The number 3 should be multiplied over both the values inside the bracket. That is 3x+6.

2. 6x+2y=8xy - In this case, we cannot add the terms with different variable.

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