### Video Transcript

Given that π₯ cubed minus 512 equals π₯ minus eight times π₯ squared plus π plus 64, find the expression for π.

If we have an expression in the form of π₯ cubed minus π¦ cubed, we can say that it can be factorized by difference of cubes, meaning we factorize it by putting it into the form of π₯ minus π¦ times π₯ squared plus π₯ times π¦ plus π¦ squared.

And hereβs what we are given. π₯ cubed minus 512 equals π₯ minus eight times π₯ squared plus π plus 64. And we need to figure out the expression for π. Just from what they gave us, we can see that π₯ is π₯ and π¦ is equal to eight. And we can even see that when we plug in π₯ into π₯ squared, so we just square it, we get π₯ squared. And then at the end, when we plug in eight for π¦ squared, eight squared is indeed 64.

Now how did they get π₯ and eight? From here. So the difference of cubes means thereβs a subtraction sign between them. And there is. And these are perfect cubes. So if we would take the cube root of each of them, so the cube root of π₯ cubed is π₯ and the cube root of 512 is eight.

So we know that our expression for π should be equal to π₯ times π¦. So we know that π₯ is π₯ and π¦ is eight. And π₯ times eight we would write as eight π₯. So the expression for π would be eight π₯. So completing everything, plugging in π, we have that π₯ cubed minus 512 can be factored by a difference of cubes to be π₯ minus eight times π₯ squared plus eight π₯ plus 64.