Algebraic Identities:
\((a + b)^2 = a^2 + 2ab + b^2\)
\((a - b)^2 = a^2 - 2ab + b^2\)
Factorisation using Identities:
Let us find the factor of the expression \(x^2 + 6x + 9\).
\(x^2 + 6x + 9 = x^2 + 2(3)(x) + 3^2\)
It looks like the identity \(a^2 + 2ab + b^2 = (a + b)^2\).
Here, \(a = x\) and \(b = 3\).
\(x^2 + 6x + 9 = (x + 3)^2\)
Thus, the factor of \(x^2 + 6x + 9\) is \((x + 3)^2\).
Factorisation by splitting the middle term:
Suppose, an algebraic expression in the form of \(ax^2 + bx + c\) cannot be factorised directly using algebraic identities. We can factore it by splitting the middle term \((bx)\) into two terms whose product is \(ac\) and whose sum is \(b\).
How it works?
Consider the quadratic expression \( ax^2 + bx + c\).
(1) Calculate \(ac\) (product of first and last term).
(2) Find two numbers whose product is \(ac\) and whose sum is \(b\).
(3) Split the middle term \(bx\) as those two numbers.
(4) Factorise by grouping.
(5) Take common factors from each group and write in factorised form.
Example:
Factorise \(x^2 + 3x + 2\).
We need two numbers whose product is \(2\) and the sum is \(3\).
The only possible numbers are \(1\) and \(2\).
\(x^2 + 3x + 2 = x^2 + x + 2x + 2\)
\(= x(x + 1) + 2(x + 1)\)
\(= (x + 1) (x + 2)\)
The factors are \((x + 1)(x + 2)\).
Simplification of rational expression:
(1) Factorise the numerator.
(2) Factorise the denominator.
(3) Cancel the common factors.
(4) Write the simplified form of the expression.
Example:
Simplify the expression \(\frac{x^2 - 7x + 12}{x^2 + x - 20}\)
Solution:
Let us factorise the numerator and denominator separately.
\(x^2 - 7x + 12 = x^2 - 3x - 4x + 12\)
\(= x(x - 3) - 4(x - 3)\)
\(= (x - 4) (x - 3)\)
\(x^2 + x - 20 = x^2 + 5x - 4x - 20\)
\(= x(x + 5) - 4(x + 5)\)
\(= (x - 4) (x + 5)\)
Now, cancel the common factors.
\(\frac{x^2 - 7x + 12}{x^2 + x - 20}\) \(=\) \(\frac{(x - 4) (x - 3)}{(x - 4) (x + 5)}\)
\(=\) \(\frac{(x - 3)}{(x + 5)}\)
Thus, \(\frac{x^2 - 7x + 12}{x^2 + x - 20}\) \(=\) \(\frac{x - 3}{x + 5}\).
Some More Identities:
\(a^2 - b^2 = (a + b)(a - b)\)
\((a + b)^3 = a^3 + 3ab^2 + 3a^2b + b^3 = a^3 + 3ab(a + b) + b^3\)
\((a - b)^3 = a^3 + 3ab^2 - 3a^2b - b^3 = a^3 - 3ab(a - b) - b^3\)
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)