Results 1 to 3 of 3

Math Help - Factorise

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    23

    Factorise

    How do i complete factorise x^3+6 in Q[x]?
    Last edited by thegarden; December 2nd 2008 at 08:24 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by thegarden View Post
    How do i complete factorise x^22 -1 in Q[x]?
    \left( x^{11} \right )^2-1=(x^{11}-1)(x^{11}+1)=p(x)\cdot q(x)


    Now by the rational roots theorem the only rational roots for each of the above binomials is \pm 1

    note that p(1)=(1^{11}-1)=0 this tells us that 1 is a root of x^{11}-1 and that (x-1) is a factor.

    You can now factor this by using long division and repeat until the rational roots theorem no longer works.

    I hope this helps.

    p.s you will find that plus or minus one are the only rational roots.

    Good luck.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi.

    That can be done with cyclotomic polynomials:
    \forall k \in \mathbb{N}-\{0\}, \prod\limits_{d|k}\Phi_{d}(X)

    Furthermore, if p is a prime number, m,r \in \mathbb{N}-\{0\} and p does'nt divides r, then:
    \Phi_{p^{m}r}(X)=\frac{\Phi_{r}(X^{p^{m}})}{\Phi_{  r}(X^{p^{m-1}})}

    As \Phi_{22}(X)=\Phi_{11.2}(X)=\frac{\Phi_{2}(X^{11})  }{\Phi_{2}(X)}=\frac{X^{11}+1}{X+1}=\sum\limits_{i  =0}^{i=10}(-1)^{i}X^{i} , and \Phi_{11}(X)=\sum\limits_{i=0}^{i=10}X^{i} we finally have:

    X^{22}-1=\Phi_{1}(X)\Phi_{2}(X)\Phi_{11}(X)\Phi_{22}(X)= (X-1)(X+1)(\sum\limits_{i=0}^{i=10}X^{i})(\sum\limits  _{i=0}^{i=10}(-1)^{i}X^{i}) ,

    and all those polynomials are irreducible in \mathbb{Q}[X].
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Factorise.
    Posted in the Algebra Forum
    Replies: 4
    Last Post: April 26th 2011, 04:07 AM
  2. Factorise
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 11th 2010, 10:48 PM
  3. factorise
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 25th 2010, 01:27 AM
  4. should this factorise?
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: February 15th 2010, 10:14 PM
  5. Factorise :/
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 6th 2010, 12:01 PM

Search Tags


/mathhelpforum @mathhelpforum