Using complete sentences, explain how you would factor completely x^9 –27.

Please Help!!

Question

Using complete sentences, explain how you would factor completely x^9 –27.

Please Help!!

turingtest · Answer

this is of the "difference of cubes" form:$$a^3-b^3=(a-b)(a^2+ab+b^2)$$so we need to identify a and b in this case, then put those numbers into this form. To do this take the cube root of each term:$$\sqrt[3]{x^9}=x^3$$$$\sqrt[3]{27}=3$$so a=x^3 and b=3
Putting this into the form above yields$$(x^3-3)(x^6+3x^3+9)$$

jamesj · Answer

Now ... by the same principle you can factor the  (x^3 - 3) term

anonymous · Answer

So i would have to do that again?

turingtest · Answer

Yeah, I was thinking about that James, but do they really think an expression containing the cubed root of 3 more simplified?$$(x-\sqrt[3]{3})(x^2+x \sqrt[3]{3}+\sqrt[3]{9})(x^6+3x^3+9)$$would be how it turns out. Ugh! ugly in my opinion.

jamesj · Answer

It's definitely more 'simple', but if you want a factorization in terms of prime polynominals over the reals (which is what is being asked, I think), what you've written is it.

anonymous · Answer

Thanks guys