Question

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

Answer 1

Recognize that 216 = 6^3

Answer 2

Then\[x^3+216=x^3+6^3\]

Answer 3

You can split off a factor of (x+6) now.

Answer 4

You can use polynomial division, or any other method that works (there are a few).

Answer 5

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,

\[(x+6)(x^2-6x+36)\]

Answer 6

You can't factorize any further unless you use complex numbers.