What is Factorisation of Trinomials (Splitting the Middle Term)?
A trinomial of the form ax² + bx + c can be factored by splitting the middle term bx into two terms whose coefficients multiply to ac and add to b.
Steps for Splitting the Middle Term (when a = 1):
- Find two numbers whose product = c and sum = b
- Rewrite bx as the sum of these two numbers times x
- Factor by grouping
Example: x² + 7x + 12
- Find numbers with product 12 and sum 7 → 3 and 4
- x² + 3x + 4x + 12
- x(x + 3) + 4(x + 3) = (x + 3)(x + 4)
What is Difference of Squares?
A difference of two squares is an expression of the form a² - b², which factors as:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Examples:
- x² - 9 = (x - 3)(x + 3)
- 4x² - 25 = (2x - 5)(2x + 5)
Important Note: Sum of squares (a² + b²) cannot be factored using real numbers.