### Theory:

What are the factors?

A factor is a number that divides the given number exactly without a remainder.
Example:
1. $$5$$ is divisible by $$1$$ and $$5$$.

2. $$20$$ is divisible by $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, and $$20$$.
From the above example, $$1$$ and $$5$$ are factors of number $$5$$.

Similarly, $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, and $$20$$ are factors of number $$20$$.

Factors are multiplied among themselves to form the original number.

$$1 \times 5 = 5$$

$$1 \times 20 = 2 \times 10 = 4 \times 5 = 20$$

We know that we can factorise a number.

But do you think it is possible to factorise an expression?

Yes, it is possible to factorise an expression. One of the common methods is expressions using identities.
Example:
1. General factorisation

Let us factorise the expression $$ab^3c$$.

We should expand the expression to find its factors.

On expansion, the expression $$ab^3c$$ becomes $$a \times b \times b \times b \times c$$.

Therefore, the factors of the expression are $$a$$, $$b$$, and $$c$$.

2. Factorisation using identities

Let us try to factorise the expression $$9 - y^2$$.

$$9 - y^2$$ can also be written as $$3^2 - y^2$$.

We know that, $$a^2 - b^2$$ $$= (a + b)(a - b)$$.

On applying the identity, we get:

$$3^2 - y^2$$ $$= (3 + y)(3 - y)$$

Thus, the factors of $$3^2 - y^2$$ are $$(3 + y)$$ and $$(3 - y)$$.
A list of common identities:

1. $$(x + a)(x + b) = x^2 + x(a + b) + ab$$

2. $$(a + b)^2 = a^2 + 2ab + b^2$$

3. $$(a - b)^2 = a^2 - 2ab + b^2$$

4. $$(a + b)(a - b) = a^2 - b^2$$