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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 \(20\).
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. \((a + b)(a - b) = a^2 - b^2\)
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