Reflection

While learning this topic, I learnt that Expansion and Factorisation requires a lot of practice. Initially this topic was very tedious and it was vey difficult for me to understand but after practice, I was able to grasp the concept and it became easier for me to solve the questions. In addition, this blog helped me to revise what was taught and definitely benefited me. It helped me clarify some doubts and definitely better allowed me to remember the concepts and fomulas.

Difference between Expansion and Factorisation

Factoring and expanding are exactly the opposite operation. To factor means to simplify the question and express the equation as a product. To expand means to multiply something out and open the brackets. 

What is Factorisation?

Factorisation is the process of expressing an algebraic expression as a product of two or more algebraic expressions. Factorisation is the reverse of expansion.

x2 + 5x+6 = x2 + 3x+ 2x+ 6
= x(x+ 3) + 2x+ 6                 (by factorising the first two terms)
= x(x+ 3) + 2(x+ 3)              (by factorising the last two terms)
= (x+ 3)(x+ 2)                     (by noting the common factor of x+ 3)

Example
Suppose we want to factorise the quadratic expression x2 −9x−22 by inspection.
We write
x2 −9x−22 = (      )(      )
and try to place the correct quantities in brackets. Clearly we will need an x in both terms:
x2 −9x−22 = (   x   )(   x  )
We want two numbers which multiply to give −22 and add to give −9. The two numbers are
−11 and +2.

x2 −9x−22 = (x−11)(x+ 2)

What is Expansion?

A quadratic expression is an expression involving a squared term, e.g., x

2 + 1, or a product term, e.g., 3xy − 2x + 1. (A linear expression such as x + 1

is obviously non-quadratic.)

This is also known as the Distributive law, where 'a groups of b and c' is the same as 'a groups of c', i.e. 'a times of b' and 'a times of c' 

Example: 

2(x - 2) = 2x - 4

Using the distributing law, 2(x - 2) = '2 groups of (x - 2) 

 Therefore, 2(x - 2) = 2x - 4. 

Quadratic expressions can also come in the form of (a + b)(c + d) . 

(a + b)(c + d) = a(c + d) + b(c + d) 

                    = ac + ad + bc + bd 

Example: 

(x + 4)(x + 5) = x(x + 5) + 4(x + 5)         (multiply the expression in the second bracket by each                                                                                term in the first bracket) 

                     = x² + 5x + 4x + 20            (Distributive law) 

                     = x² + 9x + 20                   (group like terms)