Factorise fully
1) x²y - 5yx - 6y
To factorise this expression, we can first look for the greatest common factor (GCF) among the terms. We can see that they all have a common factor of 'y'. So, we can factor out 'y' from each term:
y(x² - 5x - 6)
Now, we need to factorise the quadratic expression inside the parenthesis. We can do this by finding two numbers that multiply to -6 and add to -5. These numbers are -6 and 1:
y(x - 6)(x + 1)
So, the fully factorised form of the given expression is:
y(x - 6)(x + 1)
2) w² (a - b) - 9 (a - b)
In this expression, we can notice that both terms have a common factor of (a - b). So, we can factor out (a - b) from each term:
(a - b)(w² - 9)
Now, we can see that the expression inside the parenthesis is a difference of two squares. We can factorise it further as (w + 3)(w - 3):
(a - b)(w + 3)(w - 3)
So, the fully factorised form of the given expression is:
(a - b)(w + 3)(w - 3)
By factorising expressions fully, we can simplify them and make them easier to understand or use in further calculations. It is an essential skill in algebra and helps in solving equations, graphing functions, and finding solutions to various mathematical problems.