## 1. Introduction to Factorization

Factorisation

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)

=3x(x2−5x+4)

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

Here the term 5xy is common in the expression.

10xy+5x 3y

=(5xy×2+5xy×x2)=5xy(2+x2)

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)(n2+1+m2+1)

=(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.

3m2+mn+3mn+n2

=m(3m+n)+n(3m+n)

=(3m+n)(m+n)

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.

Example:

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.