how do you factor cubes? ex. x^3+216 ex. x^3-1

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how do you factor cubes? ex. x^3+216 ex. x^3-1

Mathematics
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Recognize that 216 = 6^3
Then\[x^3+216=x^3+6^3\]
You can split off a factor of (x+6) now.

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You can use polynomial division, or any other method that works (there are a few).
I do this...take the factor and write it out as many times as the highest power, like (x+6) (x+6) (x+6) leaving spaces. Then I ask myself, "What do I need to multiply the first one by in order to get x^3?" The answer is x^2. So I write, x^2(x+6) (x+6) (x+6) Next, when I mentally expand that first factor, I'll get x^3 +3x^2. I don't want 6x^2, so looking at the next factor I ask, "What do I have to multiply the second factor by to get rid of the 6x^2?" Well, x^2(x+6) -6x(x+6) (x+6) Now when I mentally expand the second bracket, I get -6x^2 (good) but now also, -36x. I look to the last factor and ask, "What do I have to multiply this one by in order to get rid of the -36x?" Well, multiply it by 36. Then, x^2(x+6) - 6x(x+6) + 36(x+6) Now when I expand the last one mentally, I get 36x (good) and 216 (good again). The common factor of (x+6) can be taken out from above to give, \[(x+6)(x^2-6x+36)\]
You can't factorize any further unless you use complex numbers.

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