Detailed Summary
In this section, we explore the method of factorisation by regrouping terms, which is particularly useful in algebraic expressions where no single common factor is apparent. The process begins by identifying terms that can be grouped based on shared factors. For example, in the expression 2xy + 2y + 3x + 3, we can observe that the first two terms share 2y as a common factor, while the last two terms share 3. This allows us to factor each group individually:
- 2xy + 2y = 2y(x + 1)
- 3x + 3 = 3(x + 1)
This results in the expression being rewritten as (x + 1)(2y + 3). The significance of this method lies in its utility for expressions where obvious single factors do not exist, enabling a clearer pathway to factorisation. Furthermore, the technique encourages a deeper understanding of the relationship between terms and the common factors they may contain. Through multiple examples and explanations, students can develop skills to recognise how to rearrange terms effectively for factorisation.
Example : Factorise 5xy - 10y + 15 - 5x.
Solution:
Step 1
Check if there is a common factor among all terms. There is none.
Step 2
Think of grouping. Notice that the first two terms have a common factor of 5y:
5xy - 10y = 5y(x - 2)
What about the last two terms? Observe them. If you change their order to -5x + 15, it will come out:
-5x + 15 = -5(x - 3)
Step 3
Putting (a) and (b) together,
5xy - 10y + 15 - 5x = 5y(x - 2) - 5(x - 3) = 5y(x - 2) - 5(x - 3)
The factors of (5xy - 10y + 15 - 5x) are (5y - 5)(x - 2)