## 1. Introduction to Factorization

- Books Name
- class 8 th Mathematics Book

- Publication
- ReginaTagebücher

- Course
- CBSE Class 8

- Subject
- Mathmatics

** ****Factorisation**

**Introduction to Factorization**

**Type I-Factorisation by common factor**

1. Consider the expression 3x^{3}−15x^{2}+12x.

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

(3x×x^{2})+(3x ×−5x)+(3x×4)

=3x(x^{2}−5x+4)

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

Here the term 5xy is common in the expression.

10xy+5x ^{3}y

=(5xy×2+5xy×x^{2})=5xy(2+x^{2})

**Type II-Factorisation by common binomial factor**

Consider (n^{2}+1)(m−n)+(m^{2}+1)(m−n).

Take binomial factor from each term commonly outside.

(n^{2}+1)(m−n) +(m^{2}+1)(m−n)

=(m−n)(n^{2}+1+m^{2}+1)

=(m−n)(n ^{2}+m^{2}+2)

**Type III-Factorisation by grouping**

Consider the expression 3m^{2} +mn+3mn+n^{2}.

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

3m^{2}+mn+3mn+n^{2}

=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

- Books Name
- class 8 th Mathematics Book

- Publication
- ReginaTagebücher

- Course
- CBSE Class 8

- Subject
- Mathmatics

**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 40xy^{2} /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.