We already studied factoring the algebraic expression using algebraic identities. Now, we are going to learn how to factor the algebraic expression 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:
(1) 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 \((x + 1)(x + 2)\).
(2) Factorise \(3x^2 - 10x - 8\).
We need two numbers whose product is \(3 \times 8 = 24\) and the sum is \(-10\).
The only possible numbers are \(-12\) and \(2\).
\(3x^2 - 10x - 8 = 3x^2 - 12x + 2x - 8\)
\(= 3x(x - 4) + 2(x - 4)\)
\(= (x - 4) (3x + 2)\)
The factors are \((x - 4) (3x + 2)\).