In algebra, we don’t very often use numbers (integers or decimals), we use letters as placeholders, we call these “variables”.


Take for example the very simple expression:



As you would expect, you would say this “five ecks” and it means “five times the value assigned to the variable “x””. Now because the variable can have any value, the result can also have any value. The only time that the result will be fixed is if we introduce an equality, so take the example below:



Now this time, we have changed things a little bit. “x” can now only have one value, in this case 7. The number in front of the variable, in the above case 5, is known as the “coefficient” of the variable.


Algebraic expressions sometimes have to be manipulated to make them simpler to use, here we are going to look at a reversible method called ‘expansion’. The reverse operation to ‘expansion’ is ‘factorisation’, let’s take a look at an example:



When we see an expression like this, what we have to do to expand it is to multiply everything outside the bracket by everything inside the bracket. What do we have?


  • Outside the bracket we have a single “x”, which is positive
  • Inside the bracket we have two things, a positive “x” and a negative 4

Let’s make it a little bit simpler by highlighting the different areas.



To expand this expression, we multiply “blue by green” and “blue by yellow” taking into account the fact that blue and green are both positive (as you know it is only in the case of a positive, that we don’t show the sign when it is the first variable). Once we have multiplied these entities together, we simply group them and will then have an answer.



All I have done here is to colour up the relevant bits. For example, “blue x” multiplied by “green x” gives us “pink x squared” and “blue x” multiplied by “yellow -4” gives us “Orange - 4x”


The point I’m trying to make here is that we have to make sure that when we expand, we multiply EVERYTHING outside by EVERYTHING inside, remembering to take into account that each variable has a sign.


One last thing to note before you have a go at some yourself is that if your expression once expanded seems to start with the variable with a negative sign, it usual to swap this round, if possible, so that the expression starts with a variable (or number) with a positive sign.



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