Given that 𝑥 cubed minus 512 equals 𝑥 minus eight times 𝑥 squared plus 𝑘 plus 64, find the expression for 𝑘.
If we have an expression in the form of 𝑥 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 𝑥 minus 𝑦 times 𝑥 squared plus 𝑥 times 𝑦 plus 𝑦 squared.
And here’s what we are given. 𝑥 cubed minus 512 equals 𝑥 minus eight times 𝑥 squared plus 𝑘 plus 64. And we need to figure out the expression for 𝑘. Just from what they gave us, we can see that 𝑥 is 𝑥 and 𝑦 is equal to eight. And we can even see that when we plug in 𝑥 into 𝑥 squared, so we just square it, we get 𝑥 squared. And then at the end, when we plug in eight for 𝑦 squared, eight squared is indeed 64.
Now how did they get 𝑥 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 𝑥 cubed is 𝑥 and the cube root of 512 is eight.
So we know that our expression for 𝑘 should be equal to 𝑥 times 𝑦. So we know that 𝑥 is 𝑥 and 𝑦 is eight. And 𝑥 times eight we would write as eight 𝑥. So the expression for 𝑘 would be eight 𝑥. So completing everything, plugging in 𝑘, we have that 𝑥 cubed minus 512 can be factored by a difference of cubes to be 𝑥 minus eight times 𝑥 squared plus eight 𝑥 plus 64.