Question Video: Finding an Unknown by Factorizing the Difference of two Cubes

Given that π‘₯Β³ βˆ’ 512 = (π‘₯ βˆ’ 8) (π‘₯Β² + π‘˜ + 64), find the expression for π‘˜.

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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 𝑦 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 𝑦 times π‘₯ squared plus π‘₯ times 𝑦 plus 𝑦 squared.

And here’s 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 𝑦 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 𝑦 squared, eight squared is indeed 64.

Now how did they get π‘₯ and eight? From here. So the difference of cubes means there’s 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 𝑦. So we know that π‘₯ is π‘₯ and 𝑦 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.

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