Results 1 to 6 of 6

Math Help - Finding Zeros of Cubic Polynomial

  1. #1
    Junior Member
    Joined
    Oct 2012
    From
    Houston
    Posts
    56

    Finding Zeros of Cubic Polynomial

    I am having problems factoring this cubic polynomial. I can't find how they got to the factored form on the right.

    However i was able to get to the same answer but first factored, then found possible zeros using p/q, then did long division, then used the quadratic formula. Is there a faster way to do this?

    Thanks.
    Attached Thumbnails Attached Thumbnails Finding Zeros of Cubic Polynomial-photo.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,032
    Thanks
    742

    Re: Finding Zeros of Cubic Polynomial

    Hey petenice.

    Factoring is a weird thing: in root finding a wide range of approaches are taken.

    One that is commonly taught is to "guess" one root and then factor the polynomial by dividing P(X) by (x-a) where a is the root. It's not a mechanical systematic thing unless you have previous experience or can "see" something in the polynomial itself (i.e. an easy solution).

    There is however a formula for obtaining the roots to a cubic polynomial given a cubic equation and it is more complex than the standard quadratic.

    You also have other results in algebra that give ways to factorize equations of a particular form.

    The cubic formula gives you a systematic way to get the roots and the proof shows you how the routine is constructed, but when you don't have these and try to "guess" roots then that's what happens: you make a few guesses and if you get one right then you do the long division with a polynomial and you factorize out to solve something simpler.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Finding Zeros of Cubic Polynomial

    I would use the rational roots theorem to see if a rational root exists as you did, but I would first factor out a 2 to make things a little simpler.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Sep 2012
    From
    Sweden
    Posts
    250
    Thanks
    6

    Re: Finding Zeros of Cubic Polynomial

    hello! if u intressted i can show u how to solve all the roots with rational root theorem and polynomal division!
    Last edited by Petrus; October 26th 2012 at 03:51 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Oct 2012
    From
    India
    Posts
    61
    Thanks
    3

    Re: Finding Zeros of Cubic Polynomial

    Hi, please take a look at: Cubic function - Wikipedia, the free encyclopedia, you can use some of the simpler methods there.

    Salahuddin
    Maths online
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,865
    Thanks
    744

    Re: Finding Zeros of Cubic Polynomial

    Hello, petenice!

    I don't think there is a faster way . . .


    We want to solve: . 32x^3 - 48x^2 + 12x + 2 \:=\:0

    Divide by 2: . 16x^3 - 24x^2 + 6x + 1 \:=\:0

    According to the Rational Roots Theorem,
    . . the only possible rational roots are: . \pm1,\:\pm\tfrac{1}{2},\:\pm\tfrac{1}{4},\:\pm \tfrac{1}{8},\:\pm\tfrac{1}{16}

    We find that x = \tfrac{1}{2} is a zero of the polynomial.
    . . Hence, 2x-1 is a factor.

    Using long division: . 16x^3 - 24x^2 + 6x + 1 \;=\;(2x-1)(8x^2-8x-1)


    The other two roots arise from: . 8x^2 - 8x - 1 \:=\:0

    Quadratic Formula:
    . . x \;=\;\frac{\text{-}(\text{-}8) \pm \sqrt{(\text{-}8)^2 - 4(8)(\text{-}1)}}{2(8)} \;=\;\frac{8 \pm\sqrt{96}}{16} \;=\; \frac{8 \pm4\sqrt{6}}{16} \;=\;\frac{2\pm\sqrt{6}}{4}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding zeros of a polynomial
    Posted in the Algebra Forum
    Replies: 4
    Last Post: June 25th 2012, 06:11 PM
  2. More Finding Zeros of a Polynomial
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: February 22nd 2010, 01:36 AM
  3. Finding zeros of a polynomial
    Posted in the Algebra Forum
    Replies: 4
    Last Post: July 7th 2009, 09:13 AM
  4. Finding zeros of a polynomial
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: November 20th 2008, 06:11 PM
  5. Finding Zeros in a Polynomial
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 8th 2008, 06:24 PM

Search Tags


/mathhelpforum @mathhelpforum