Factors:

We are aware that a natural number can be expressed as the product of other natural numbers. As
20 = 2 x 10 = 4 x 5
Thus, 1, 2, 4, 5, 10, 20 are the factors of 20. Of these 2 and 5 are prime factors of 20.

Similarly, there are factors of algebraic expressions as,
5ab = 5 x a x b where 5, a and b are prime factors of 5ab.

Factorisation:

When we factorise an algebraic expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.

Example: 12xy is already in factors but we write it as 3 x 2 x 2 x X x Y. thus factors are 2, 3, x, y.

Factorisation by Common Factors:

As the term common itself suggest that it must be some similar pattern.

A systematic way of factorising an expression is the common factor method. It consists of three steps:

• Write each term of the expression as a product of irreducible factors
• Look for and separate the common factors and
• Combine the remaining factors in each term in accordance with the distributive law.

As example: We have to factorise 3x + 9.
first let’s factorise 3x = 3 x X and 9 = 3 x 3
similar pattern (say common) among these factors of both are 3
thus we can write 3x + 9 = 3 x X + 3 x 3 = 3(x + 3)

Factorisation by regrouping terms:

Rearranging the expression allows us to form groups leading to factorisation. This is regrouping.

Let us write (2xy + 2y) in the factor form:
2xy + 2y = (2 × x × y) + (2 × y)
= (2 × y × x) + (2 × y × 1)
= (2y × x) + (2y × 1) = 2y (x + 1)

Factorisation using identities:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• (a + b) (a – b) = a² – b²

Factors of the product form

• (x + a).(x + b) = x² + (a + b) x + ab

Division:

We already know that division is the inverse of multiplication for numerals. It is also applicable to the division of algebraic expressions.

Division of a monomial by another monomial:

We use the simple method for division. The division of numerical is same as we divide but for the monomials, we simply subtract the power of denominator from the numerator.

Example: \(\frac{21x^{4}}{7x}\) = 3x²

Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial):

By taking the common factor we divide it.
Just an example: (7x² + 14x) ÷ (x + 2)
(7x² + 14x) = (7 × x × x) + (2 × 7 × x) = 7x(x + 2)
thus, (7x² + 14x) ÷ (x + 2)
= \(\frac{7x(x + 2)}{x + 2}\) = 7x

Examples related to factorisation:

1. Factorise 12a²b + 15ab².
   Solution:
   As, 12a²b = 2 × 2 × 3 × a × a × b
   and, 15ab² = 3 × 5 × a × b × b
   by seeing the two terms we find that these two terms have 3, a and b as common factors.
   12a²b + 15ab² = (3 × a × b × 2 × 2 × a) + (3 × a × b × 5 × b)
   = 3 × a × b × [(2 × 2 × a) + (5 × b)]
   = 3ab × (4a + 5b)
   = 3ab (4a + 5b)
2. Factorise 6xy – 4y + 6 – 9x.
   Solution:
   6xy – 4y = 2y (3x – 2)
   –9x + 6 = –3 (3x) + 3 (2)
   Taking them together,
   6xy – 4y + 6 – 9x = 6xy – 4y – 9x + 6
   = 2y (3x – 2) – 3 (3x – 2)
   = (3x – 2) (2y – 3)
   thus factors of (6xy – 4y + 6 – 9 x) are (3x – 2) and (2y – 3).