Factorise fully 𝑎 cubed plus 𝑏 cubed plus 𝑎 plus 𝑏.
Our first step is to recall that the sum of two cubes, 𝑎 cubed plus 𝑏 cubed, can be written as 𝑎 plus 𝑏 multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. This means that the first two terms in our expression can be written in the same way. Bringing down the last two terms leaves us with 𝑎 plus 𝑏 multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared plus 𝑎 plus 𝑏.
If we put the last two terms, 𝑎 plus 𝑏, into brackets, we notice that we have an 𝑎 plus 𝑏 in the first term and an 𝑎 plus 𝑏 in the second term. This means that this can be factorised out. 𝑎 plus 𝑏 is a common term. The first 𝑎 plus 𝑏 is being multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. And the second 𝑎 plus 𝑏 is being multiplied by one. This leaves us with 𝑎 plus 𝑏 multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared plus one. This is the fully factorised version of 𝑎 cubed plus 𝑏 cubed plus 𝑎 plus 𝑏.
We can check that our answer is correct by expanding the two parentheses or brackets. One way to do this is using the grid methods. We multiply every term in the first parenthesis by every term in the second parenthesis. 𝑎 multiplied by 𝑎 squared is equal to 𝑎 cubed. 𝑎 multiplied by negative 𝑎𝑏 is equal to negative 𝑎 squared 𝑏. 𝑎 multiplied by 𝑏 squared is equal to 𝑎𝑏 squared. And finally in the first column, 𝑎 multiplied by one is equal to 𝑎.
In the second column, multiplying 𝑏 by 𝑎 squared gives us 𝑎 squared 𝑏. Multiplying 𝑏 by negative 𝑎𝑏 gives us negative 𝑎𝑏 squared. 𝑏 multiplied by 𝑏 squared is equal to 𝑏 cubed. And finally, 𝑏 multiplied by one is equal to 𝑏. The two 𝑎 squared 𝑏 terms cancel. Likewise, the two 𝑎𝑏 squared terms cancel. We are left with 𝑎 cubed plus 𝑏 cubed plus 𝑎 plus 𝑏 which is the same as the expression we started with. Therefore, our factorised version is correct.