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Common quadratic factor

  1. Jan 27, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the value of m and n, where m and n are integer, so that P(x) = x3 + mx2 – nx - 3m and
    Q(x) = x3 + (m – 2) x2 –nx – 3n have common quadratic factor.


    2. Relevant equations



    3. The attempt at a solution
    Is m = n = 0 one of the solution?

    If m = n = 0,then :
    P(x) = x3 = x2 (x)

    Q(x) = x3 - 2x2 = x2 (x-2)

    Can I say that they have common quadratic factor, which is x2 ?

    The point is : Am I right to say x3 can be be factorized to x2 x or even (x) (x) (x) ?

    Thanks
     
  2. jcsd
  3. Jan 28, 2010 #2

    Mark44

    Staff: Mentor

    That works, but there might be another solution with both functions having a quadratic factor of x2 + bx + c. I've filled up a couple of pieces of paper without finding it, though.
     
  4. Jan 28, 2010 #3

    vela

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    m=5 and n=3 works.
     
  5. Jan 28, 2010 #4

    vela

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    If p(x) and q(x) have a common quadratic factor, they can be written

    [tex]p(x) = (x+a)(x^2+cx+d)[/tex]
    [tex]q(x) = (x+b)(x^2+cx+d)[/tex]

    so that

    [tex]p(x)-q(x) = (a-b)(x^2+cx+d)[/tex].

    Try plugging the given polynomials into the LHS and match coefficients to determine a, b, c, d, m, and n.
     
  6. Jan 28, 2010 #5
    Hi Mark and vela

    Thanksssss !!
     
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