### Video Transcript

Expand and simplify [π] cubed minus nine π cubed plus ππ multiplied by π minus nine π, then factorise the result.

So if we take a look at this question, it actually has two parts. First, weβll need to expand and simplify and in the second part, we need to actually factorise. So weβre gonna begin by expanding and simplifying. So when we first expand the parentheses, weβre gonna get π squared π. So then our second term is gonna be negative nine ππ squared. Thatβs cause if you have ππ multiplied by negative nine π, you get negative nine ππ squared.

So now, well we canβt actually simplify the expression any further cause we got no like terms. But what we can do is actually rearrange it so that we have common factors. So now what Iβve done is that Iβve actually rearranged it so that actually we have two terms next to each other β thatβs small groups β and actually have common factors cause what weβre gonna do now is factor it using a method of grouping.

And to do that, Iβm actually gonna factor each two terms separately. So first of all, we got π cubed plus π squared π. So therefore, we have π squared outside the parentheses and inside the parentheses we have π plus π and thatβs because π squared multiplied by π gives us π cubed and π squared multiplied by π gives us π squared π.

And then, for our next two terms or our next group, we have negative nine π cubed minus nine ππ squared. So therefore, a common factor of both of these is negative nine π squared, which means that inside the parentheses weβre gonna have π plus π and thatβs because negative nine π squared multiplied by π is negative nine π cubed and negative nine π squared multiplied by π is negative nine ππ squared.

So now, what weβre gonna do is a quick check to see if weβve done the first stage correctly. And we have because actually we have got the same factor in both of our parentheses because π plus π is the same as π plus π. And actually if you havenβt got the same factor in each parenthesis and then you need to check to make sure you have done the rest of it correctly because if you havenβt got that, then you wonβt be able to use this method.

So therefore, what we have now is π squared minus nine π squared. And thatβs because we took the factors that are outside of our parentheses and made that into one of our factors. And then, this is multiplied by π plus π which is what was in both parentheses. So now, we actually have the difference of two squares and thatβs because π squared is a squared term. And weβve got nine π squared which is also a squared term because nine is a squared number and π is obviously squared. And then, we have a negative between them.

And when we have the difference of two squares, we can actually factor this in a special way. And when we do that, we get π plus three π multiplied by π minus three π then multiplied by π plus π.

And we actually found that because the roots of π squared is equal to π. So π multiplied by π is π squared. And the root of nine π squared is three π because three π multiplied by three π gives us nine π squared. And we have negative and positive. And thatβs because when we actually expand the parentheses, what weβd actually be left with is two terms that are actually gonna cancel each other out because what weβd get is a positive three ππ and then minus three ππ. So they cancel out.

So therefore, we can say that if we expand and simplify π cubed minus nine π cubed plus ππ multiplied by π minus nine π, then the fully factorised result is π plus three π multiplied by π minus three π multiplied by π plus π.