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 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
- 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 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.