Factorise fully 𝑥 to the power of five minus 𝑥 cubed minus 𝑥 squared plus one.
In order to factorise this expression, we’ll firstly separate the first two terms and the last two terms. The first two terms have a common factor of 𝑥 cubed. Therefore, we can write this outside of the parenthesis. Inside the parenthesis is 𝑥 squared minus one. As 𝑥 cubed multiplied by 𝑥 squared is equal to 𝑥 to the power of five. And 𝑥 cubed multiplied by negative one is negative 𝑥 cubed.
We can factorise negative one out of the last two terms. This also leaves us with 𝑥 squared minus one inside of the parenthesis. As negative one multiplied by 𝑥 squared is negative 𝑥 squared. And negative one multiplied by negative one is equal to one.
At this stage, we notice that both parts of our expression have the term 𝑥 squared minus one. We can, therefore, factorise this out. The parts to the left in the second parenthesis are 𝑥 cubed minus one. 𝑥 squared minus one is the difference of two squares. And 𝑥 cubed minus one is the difference of two cubes. We can, therefore, use our rules for the difference of two squares and difference of two cubes to factorise both of these parts.
The difference of two squares states that 𝑎 squared minus 𝑏 squared is equal to 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏. In our case, 𝑎 is equal to 𝑥 and 𝑏 is equal to one as the square root of 𝑥 squared is 𝑥 and the square root of one is one. This means that 𝑥 squared minus one can be written as 𝑥 plus one multiplied by 𝑥 minus one.
The difference of two cubes, 𝑎 cubed minus 𝑏 cubed, can be written as 𝑎 minus 𝑏 multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared. In our case, once again, 𝑎 is equal to 𝑥 and 𝑏 is equal to one as the cube root of 𝑥 cubed is 𝑥 and the cube root of one is equal to one. This means that we can write 𝑥 cubed minus one as 𝑥 minus one multiplied by 𝑥 squared plus 𝑥 plus one.
We notice that our middle two parentheses are the same. 𝑥 minus one multiplied by 𝑥 minus one can be written as 𝑥 minus one squared. This means that our fully factorised expression is 𝑥 plus one multiplied by 𝑥 minus one squared multiplied by 𝑥 squared plus 𝑥 plus one. These three brackets, or parentheses, can be written in any order as multiplication is commutative.