Factorise fully: 𝑥 cubed plus four 𝑥 squared minus 𝑥 minus four.
Okay so we’re looking to actually factorise this expression. And in order to do so, what we’re actually gonna use is a method called grouping. And to actually factorise by grouping, what we’re actually gonna do is actually split our expression into two groups. So we’ve got the first two terms and the second two terms and we’re actually gonna factorise them in that way first.
So what we’re gonna start with our is first group, well we mean our first two terms. So we’ve got 𝑥 cubed plus four 𝑥 squared. Well we know that 𝑥 squared is actually gonna be a factor of both 𝑥 cubed and four 𝑥 squared. So this is gonna be outside the parentheses. And then the first term inside the parentheses is gonna be 𝑥. And that’s because 𝑥 squared multiplied by 𝑥 gives us 𝑥 cubed. And the second term is gonna be positive four. And that’s because 𝑥 squared multiplied by positive four gives us positive four 𝑥 squared.
Okay, great! So that’s the first two terms or the first group actually factorised. So now what we do is we actually move on to the second two terms or our second group. And we’ve got negative 𝑥 minus four. Well now the only factor of these two terms is actually negative one. So this is gonna go outside the parentheses. And then inside the parentheses, we’re gonna have, first of all, 𝑥. And that’s because negative one multiplied by 𝑥 gives us negative 𝑥. And then we’re gonna have positive four. And that’s because negative one multiplied by positive four gives us negative four.
Great! What we can also do to check they’re actually correct is have a look at what factor we have in each of the parentheses. And we’ve actually got the same factor, which is 𝑥 plus four. And that way, we actually know where you’ve done the first stage correctly. Because in order to use this method, we need to actually have the same factor in each of our parentheses. And if we didn’t, we’d know that we’d done something wrong. So we’d actually check through what we’d already done.
So now what we can do is actually rewrite our expression again. So we’ve got 𝑥 squared minus one because these are the two terms that are actually outside the parentheses. That’s our first factor. And our second factor is 𝑥 plus four because that was the actual value that’s in each of our parentheses. Okay, great! So have we actually finished here? Well, if we look back at the question, it says factorise fully. So it says right! In that case, what we need to do is see if we can actually factorise our expression any further.
And we actually can. And that’s because in our first parentheses, what we actually have is the difference of two squares, because what we have is a square because 𝑥 squared is a square and then we have another square, which is one. So it’s a square number. And then it’s negative in between. So we’ve got 𝑥 squared minus one. Okay, but if it’s a difference of two squares, what does this mean? Well what it means is that actually we can factorise it as 𝑥 plus one multiplied by 𝑥 minus one.
And we’re actually factorising any difference of two squares that has the same form. What I’m gonna do now is show you why it works. So now, to show you how it works, what I’m gonna do is actually expand 𝑥 plus one multiplied by 𝑥 minus one. So first of all, we have 𝑥 multiplied by 𝑥, which gives us 𝑥 squared. Then we have 𝑥 multiplied by negative one, which gives us negative 𝑥. And then we’ve got positive one multiplied by 𝑥, which gives us positive 𝑥. And then finally, we have positive one multiplied by negative one, which gives us negative one. So we’ve got 𝑥 squared minus 𝑥 plus 𝑥 minus one.
Now because each of the factors was actually a different sign, so we have positive one and negative one, this means that actually we’re gonna have two terms that are gonna cancel out, because we’ve got negative 𝑥 and positive 𝑥. So we’ve got negative 𝑥 plus 𝑥, which is just gonna be zero. So they cancel out, which leaves us with 𝑥 squared minus one, which is what we had in our parentheses. So therefore, what we can say is if you factorise fully 𝑥 cubed plus four 𝑥 squared minus 𝑥 minus four, you get 𝑥 plus one multiplied by 𝑥 minus one multiplied by 𝑥 plus four. And we found that using the grouping method.