Reflections

When I first started learning these chapters, I found the questions easy, but soon the equations started to become more complicated and challenging, and thus I did not fully understand how to expand and factorise them. I found factorisation much harder compared to expansion as there were so many methods to use for different types of equations, such as grouping, and special identities, which I found hard to memorize. But after much practice, I can now solve questions like these without much difficulty. In addition, creating this blog has truly helped me understand expansion and factorisation more clearly.

Difference Between Expansion and Factorisation

Expansion is the reversed process of factorisation. Expanding an equation means to get rid of the brackets, and so, the values outside the brackets will be multiplied to each of the values inside the bracket individually. On the other hand, factorisation means to put in the brackets, in which, the common factor is factored out, and the rest of the values will remain in a bracket.

Factorisation

Factorisation is the process of expressing an algebraic expression as a product of its factors.
1.       Factorisation by a common factor

Example 1:

       5x + 15 = 5( x + 3)

As 5 and 15 have the common factor of 5, you can factor the out the 5, and place it in front of the bracket. There, you get your answer of 5(x+3)

 

2. Special identities

a² + 2ab +b²= (a+b)²

a² - 2ab +b² = (a-b)²

a² + b² = (a+b)(a-b)

Example 2:

x²+ 12x + 36 = (a + b) ²                                               a = x

                   = (x + 6) ²                                               b = 6

 

Note:  both ‘x²’ and ‘36’ have to be perfect squares, and the highlighted sign has to be a positive sign.

3. Cross method

x² + x – 2 = ( x + 2)(x – 1)

 

In the first 2 columns, the two rows must be multiplied and the product should be what is in the third row. After that, cross multiply as seen in the picture, and add the two products together in the third column, third row.  Your answers are in the two brackets.

4. Grouping

Xy+ 4x + 3y + 12 = (xy + 4x) + (3y +12)

                           =x(y + 4) + 3(y + 4)

              =(y + 4)(x + 3)

Group them into two groups with the same factors, and factorise the common factor outside the brackets. The common factor outside both brackets will then be grouped together in a bracket. There, you get your answer.
 

Expansion

 

Expansion is the process of removing the brackets in an expression and multiplying them term by term.

 

Example 1:

Take 4 and multiply it into every term inside the bracket, which in this case is 2x and 3. The product of 4 and 2x is 8x, and the product of 4 and 3 is 12.

After that, place your answers together, and you will get 8x + 12

 

 

Example 2:

 

Take the numbers in the first bracket and multiply them with every term in the second bracket individually. So, take x times y and the product will be xy , and multiply x to (-2) , the product being -(2x)

Later, multiply 3 to y , and the product is 3y and multiply 3 to 2 , and the product is 6 .    
So the answer will be xy - 2x + 3y -6
 

 

 

 

 

Both examples make use of the distributive law, where “a times b and c” is the same as “a times b” and “a times c”

 

Special identities can also be used for expansion

1. (a+b)² = a² + 2ab +b²

2. (a-b)² = a² - 2ab +b²

3. (a+b)(a-b) = a² + b²

Example 3:

(2x + 3)² = a² +2ab + b²                                                 a= 2x                                                         

              = (2x)² + 2(2x)(3) + 3²                                     b= 3

              = 4x² +12x + 9

 

Example 4:

(a + 2b)(a-2b) = a² + b²                                                 a= a

                      = a² - (2b)²                                              b= 2b

                      = a²-4b²