Results 1 to 10 of 10

Math Help - [Factorization] Factorization of polynomials

  1. #1
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71

    [Factorization] Factorization of polynomials

    I need help on how to factorize the following w/o calculator if possible.
    x^4+5x^3+2x^2+x-3=0
    I've attempted to factor with whole numbers with no avail.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,206
    Thanks
    1789
    Quote Originally Posted by Cthul View Post
    I need help on how to factorize the following w/o calculator if possible.
    x^4+5x^3+2x^2+x-3=0
    I've attempted to factor with whole numbers with no avail.
    By the "rational root theorem", the only possible rational roots of that equation would be factors of 3: 1, -1, 3, and -3. Since none of those satisfy the equation, there are no rational roots. The polynomial cannot be factored with integer coefficients.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71
    Okay.
    I've checked the answers and the factor of it is.
    (x^2+x+1)(x^2+4x-3)=0
    For the solution to be true
    x=-2 \pm \sqrt {7}
    From what I know, via the answers.
    (x^2+x+1)=0
    Has no solutions because
    \Delta=-3
    \Delta<0
    Then there are no solutions.

    So, is there no way to do this mentally or by hand? Assuming I am factorizing over irrational numbers.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Cthul View Post
    Okay.
    I've checked the answers and the factor of it is.
    (x^2+x+1)(x^2+4x-3)=0
    For the solution to be true
    x=-2 \pm \sqrt {7}
    From what I know, via the answers.
    (x^2+x+1)=0
    Has no solutions because
    \Delta=-3
    \Delta<0
    Then there are no solutions.

    So, is there no way to do this mentally or by hand? Assuming I am factorizing over irrational numbers.
    HOI reasonably assumed that you wanted linear factors.

    Since linear factors involving integers are not possible, it's natural to then try to factor it as a product of two quadratics:

    (x^2 + ax + b)(x^2 + cx + d)

    and then see if you can find integer values of a, b, c and d.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71
    Oh, I see. Thanks.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71
    I'm still stuck on this equation. I tried factorizing by identical equations but I end up with too many variables, is there really no way to factor this? I want to know how to factorize this.

    (My attempt to factorize)
    The identity?
    (x^2+ax+b)(x^2+cx+d)=0
    Expanded.
    x^4+ax^3+cx^3+acx^2+bx^2+dx^2+adx+bcx+bd=0
    Common Factor.
    x^4+x^3(a+c)+x^2(ac+b+d)+x(ad+bc)+bd=0

    So
    5=a+c
    2=ac+b+d
    1=ad+bc
    -3=bd

    Original Equation:
    x^4+5x^3+2x^2+x-3=0
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71
    So does anyone have a method to solve this?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Cthul View Post
    So does anyone have a method to solve this?
    Are you expected to solve that specific sort of quartic equation without a calculator? Has it come from a section of work where using a calculator is not permitted?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member Cthul's Avatar
    Joined
    Mar 2010
    Posts
    71
    It's an exercise from the book, and the section specified that calculators are not to be used.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member Bacterius's Avatar
    Joined
    Nov 2009
    From
    Wellington
    Posts
    927
    Quote Originally Posted by Cthul View Post
    It's an exercise from the book, and the section specified that calculators are not to be used.
    Hello,
    you might want to solve the system for a, b, c and d, thus recovering the factorized expression of the polynomial and solving it in the standard way.

    However I don't know if it is possible to solve this system. You have four unknowns and four equations, but how to solve it I have no idea yet (haven't really looked). Try to work out some algebra around the system and see what you can do ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. qr factorization
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 23rd 2010, 06:46 PM
  2. Proving unique factorization for polynomials
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: July 3rd 2010, 10:44 PM
  3. Replies: 2
    Last Post: March 4th 2010, 02:49 AM
  4. Factorization help
    Posted in the Algebra Forum
    Replies: 3
    Last Post: January 30th 2010, 03:21 PM
  5. Delta and factorization of polynomials?
    Posted in the Pre-Calculus Forum
    Replies: 13
    Last Post: September 22nd 2009, 11:15 AM

Search Tags


/mathhelpforum @mathhelpforum