# 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}= a^{2}+ 2ab + b^{2} - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b) (a – b) = a
^{2}– b^{2}

**Factors of the product form**

- (x + a).(x + b) = x
^{2}+ (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:** \(\frac{21x^{4}}{7x}\) = 3x^{2}

**Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial):**

By taking the common factor we divide it.

Just an example: (7x^{2} + 14x) ÷ (x + 2)

(7x^{2} + 14x) = (7 × x × x) + (2 × 7 × x) = 7x(x + 2)

thus, (7x^{2} + 14x) ÷ (x + 2)

= \(\frac{7x(x + 2)}{x + 2}\) = 7x

**Examples related to factorisation:**

.*Factorise 12a*^{2}b + 15ab^{2}

**Solution:**As, 12a

^{2}b = 2 × 2 × 3 × a × a × b

and, 15ab^{2}= 3 × 5 × a × b × b

by seeing the two terms we find that these two terms have 3, a and b as common factors.

12a^{2}b + 15ab^{2}= (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)6xy – 4y = 2y (3x – 2)*Factorise 6xy – 4y + 6 – 9x.*Solution:

–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).

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