# Math Help - Factorising Polynomial over the real number field

1. ## Factorising Polynomial over the real number field

hey guys I was just wondering whether I'm doing this right
for example the question asks to factor $z^6+8$ over the real number field

Heres what I did
$z^6+8=(z^2)^3+2^3$
$=(z^2+2)(z^4-2z^2+4)$

but when I checked the answers they got
$z^6+8=(z^2+2)(z^2+\sqrt6z+2)(z^2-\sqrt6z+2)$
they got this by pairing up the roots

which is more factored than my expression, the factors I got are not factorised enough right?, I have tried difference of 2 squares, yet they don't give me the same as the answer as they are complex
$=(z^2+2)(z^4-2z^4+4)$
$=(z^2+2)((z^2+1)^2-3i^2)$
$=(z^2+2)(z^2+1-\sqrt3i)(z^2+1+\sqrt3i)$

• Is the method I used originally wrong, since it is not fully factored?
• How would I be able to factorise it to the right answer (eg. provided by the textbook) by just continuing with the factors I originally got, without using the pairing up of roots method the text book used. Is it possible?

Thanks
Aonin

2. Difference of squares? Sure they do. Just rewrite a little:

$z^{4} - 2\cdot z^{2} + 4 = z^{4} + 4 - 2\cdot z^{2} = (z^{2} + 2)^{2} - 2\cdot z^{2} - 4\cdot z^{2} = (z^{2} + 2)^{2} - 6\cdot z^{2}$

You're just not used to completing the square on the usually-outside pieces. Add it to your bag of tricks. You might need it again.

so by completing the square on the outside pieces, its possible to factor any polynomial that couldn't be factored otherwise..or is this a generalisation?

4. Didn't you ever demonstarte the development of the Quadratic Formula?

$x^{4} + b^{2}x^{2} + c^{2} =$

$\left(x^{4} + c^{2}\right) + b^{2}x^{2} =$

$(x^{2}+c)^{2} + b^{2}x^{2} - 2cx^{2} =$

$(x^{2}+c)^{2} + \left(b^{2} - 2c\right)x^{2} =$

$(x^{2}+c)^{2} - \left(2c - b^{2}\right)x^{2} =$

That's an important point. $2c - b^2 > 0$ I'm guessing you'll just get more imaginaries otherwise.

$(x^{2} + c + \sqrt{2c - b^{2}}|x|)\cdot (x^{2} + c - \sqrt{2c - b^{2}}|x|)$

I'm going to stick my neck out and say that you may have a chance of finding this result in a book on quartic equations by Orson Pratt in the middle 19th century. It's not my favorite result. I'm not very happy about the absolute values, either.

I think, if you force a few things into this form, something may occasionally come of it.