Results 1 to 8 of 8

Thread: Factor quartic equation

  1. #1
    Junior Member
    Joined
    Sep 2016
    From
    Muskoka
    Posts
    58

    Factor quartic equation

    How would I factor x^4+3x^2+6x+10?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,069
    Thanks
    412

    Re: Factor quartic equation

    (x^2 - 2x + 5)(?)
    Do some work!!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2016
    From
    Muskoka
    Posts
    58

    Re: Factor quartic equation

    x^2+2x+2?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    6,359
    Thanks
    2732

    Re: Factor quartic equation

    Quote Originally Posted by Ilikebugs View Post
    x^2+2x+2?
    correct
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    227
    Thanks
    108

    Re: Factor quartic equation

    Quote Originally Posted by DenisB View Post
    (x^2 - 2x + 5)(?)
    Do some work!!
    Which work? Where did you get that trinomial?

    Quote Originally Posted by Ilikebugs View Post
    x^2+2x+2?
    Same question. Where did you get that trinomial?

    If you already had the trinomial from DenisB, I suppose you might have gotten it by doing polynomial division.

    For instance, it does not factor by grouping.

    So, if someone were to use the Rational Root Theorem, they would see that it has no rational roots. So, either it factors as
    two binomials with integer coefficients or it doesn't factor (over integer coefficients).

    But how did the initial quadratic factor come about from DenisB?
    Last edited by greg1313; Feb 1st 2018 at 12:10 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,069
    Thanks
    412

    Re: Factor quartic equation

    Quote Originally Posted by greg1313 View Post
    But how did the initial quadratic factor come about from DenisB?
    Ain't telling ya!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    22,150
    Thanks
    3042
    Awards
    1

    Re: Factor quartic equation

    Quote Originally Posted by Ilikebugs View Post
    How would I factor x^4+3x^2+6x+10?
    $\large(x^2-2x+5)(x^2+2x+2)=~?$
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Apr 2005
    Posts
    20,045
    Thanks
    3173

    Re: Factor quartic equation

    Well, you can't just stare at a polynomial and expect to suddenly see its factors! You have to try things and do some calculations. First, when we talk about "factoring" a polynomial, with integer coefficients we generally mean into lower degree polynomials with integer coefficients. Here the polynomial is $x^4+ 3x^2+ 6x+ 10$. 10 factors as (1)(10) or (2)(5).

    So we might try $(x^2+ ax+ 1)(x^2+ bx+ 10)= x^2(x^2+ bx+ 10)+ax(x^2+ bx+ 10)+ 1(x^2+ bx+ 10)= x^4+ (a+ b)x^3+ (10+ ab+ 1)x^2+ (10a+ b)x+ 10$. We want that to equal to $x^4+ 3x^2+ 6x+ 10$ so we must have a+ b= 0, 11+ ab= 3, and 10a+ b= 6. From a+ b= 0, b= -a, of course. Then 10a+ b= 10a- a= 9a= 6 so that a= 2/3 not an integer! Further, if a= 2/3, b must equal -2/3 so 11+ ab becomes 11- 4/9= 95/9, not 3.

    If we try $(x^2+ ax+ 2)(x^2+ bx+ 5)= x^2(x^2+ bx+ 5)+ ax(x^2+ bx+ 5)+ 2(x^2+ bx+ 5)= x^4+ bx^3+ 5x^2+ ax^3+ abx^2+ 5ax+ 2x^2+ 2bx+ 10= x^4+ (a+b)x^3+ (5+ab+ 2)x^2+ (5a+ 2b)x+ 10$. We want that to equal to $x^4+ 3x^2+ 6x+ 10$ so we must have a+ b= 0, 5+ ab= 3, and 5a+2b= 6. Again, a+ b= 0 gives b= -a so 5a+ 2b= 5a- 2a= 3a= 6 so a= 2 and b= -2. Then 7+ ab= 7- 4= 3!

    Yes, $(x^2+ 2x+ 2)(x^2- 2x+ 5)= x^4+ 3x^w+ 6x+ 10$.

    Of course, there are other possibilities: $(x^2+ ax- 1)(x^2+ bx- 10)$ or $(x^2+ ax- 2)(x^2+ bx- 5)$ or a linear term times a cubic but fortunately we don't need to look at those.
    Last edited by HallsofIvy; Feb 1st 2018 at 06:23 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: Jul 25th 2014, 06:56 AM
  2. Factoring Quartic equation
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Apr 6th 2010, 09:07 AM
  3. Quartic Equation with reals
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: Oct 1st 2009, 11:33 PM
  4. Solving a quartic equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Jan 15th 2009, 01:21 PM
  5. Quartic Equation
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 8th 2008, 03:57 PM

/mathhelpforum @mathhelpforum