When an expression is the product of two or more expressions, then each of the expressions is called a factor of the given expression.
The process of writing a given expression as the product of two or more factors is called factorization.
The greatest common factor of two or more monomials is the product of the greatest common factors of the numerical coefficients and the common letters with smallest powers.
When a common monomial factor occurs in each term of an algebraic expression, then it can be expressed as a product of the greatest common factor of its terms and quotient of the given expression by the greatest common factor of its terms.
When a binomial is a common factor, we write the given expression as the product of this binomial and the quotient of the given expression by this binomial.
If the given expression is the difference of two squares, then to factorize it, we use the formula (a2 – b2) = (a + b) (a – b)
If the given expression is a complete square, we use one of the following formulae to factorize it:
- a2 + 2ab + b2 = (a + b)2 = (a + b)(a + b)
- a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b)
For factorisation of the form (x2 + px + q), we find two numbers a and b such that (a + b) = p and ab = q, then x2 + px + q = x2 + (a + b)x + ab = (x + a) (x + b).
In case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
In case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the division polynomial. Instead, we factorise both the polynomial and cancel their common factors.
In the case of division of algebraic expression, we have Dividend = Divisor × Quotient + Remainder.
Factors of Natural Numbers
A number, when written as a product of its prime factors, is said to be in the prime factor form. Similarly, we can express algebraic expressions as products of their factors.
Factors of Algebraic Expressions
An irreducible factor is one which cannot be expressed further as a product of factors.
What is Factorisation?
When we factorise an algebraic expression, we write it as a product of irreducible factors. These factors may be numbers, algebraic variables or algebraic expressions.
Method of Common Factors
We factorise each term of the given algebraic expression as a product of irreducible factors and separate the common factors. Then, we combine the remaining factors in each term using the distributive law.
Factorisation By Regrouping Terms
Sometimes it so happens that all the terms in a given algebraic expression do not have a common factor; but the terms can be grouped in such a manner that all the terms in each group have a common factor. In doing so, we get a common factor across all the groups formed. This leads to the required factorisation of the given algebraic expression.
Factorisation Using Identities
The following identities prove to be quite helpful in factorisation of an algebraic expression:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a + b) (a – b) = a2 – b2
Factors of the Form (x + a) (x + b)
(x + a) (x + b) = x2 + (a + b) x + ab
To factorise an algebraic expression of the type x2 + px + q, we find two factors a and b of q such that ab = q and a + b = p
Then, the given expression becomes
x2 + (a + b) x + ab = x2 + ax + bx + ab = x (x + a) + b (x + b) = (x + a) (x + b) which are the required factors.
Division of Algebraic Expressions
Here, we shall divide one algebraic expression by another.
Division of a Monomial by Another Monomial
We shall factorise the numerator and denominator into irreducible factors and cancel out the common factors from the numerator and the denominator.
Division of a Polynomial by a Monomial
We divide each term of the polynomial in the numerator by the monomial in the denominator.
Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial)
We factorise the algebraic expressions in the numerator and the denominator into irreducible factors and cancel the common factors from the numerator and the denominator.
Rules to be Followed You Find The Errors
- Coefficient 1 of a term is usually not written. But while adding like terms, we should include it in the sum.
- When we are going to substitute a negative value, we should remember to make use of brackets.
- When we have to multiply an expression enclosed within a bracket by a constant or a variable outside, we should multiply each term of the expression by that constant or variable.
- When we have to square a polynomial, we should square the numerical coefficient and each factor.
- When we have to divide a polynomial by a monomial, we should divide each term of the polynomial in the numerator by the monomial in the denominator.