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Polynomials of different degrees and a related monomial

  1. Mar 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f, g be nonzero polynomials with deg (f) [tex]\geq[/tex] deg (g). Show that there
    is a unique monomial bx[tex]^{k}[/tex] where deg(f(x) - bx[tex]^{k}[/tex]g(x)) < deg (f).

    2. Relevant equations

    see above

    3. The attempt at a solution

    I define polynomials f and g, with deg(f) = n and deg (g) = m and n[tex]\geq[/tex]m
    and let the monomial be h(x) so h(x)g(x) = l(x) and using the theorem deg(h(x)g(x)) = deg(h(x)+g(x)) and therefore deg l(x) = k + m. so overall i have to find deg(f(x)-l(x)) but this is equal to f(x) not less than f(x), how do I show its less than f(x).
     
  2. jcsd
  3. Mar 24, 2009 #2

    epenguin

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    Re: polynomials

    In other words you have a polynomial f = anxn + an-1xn-1 + ... , and you have another one g = cmxm + cm-1xm-1 + ...
    where n > m .

    And you are asked if you can find a bxk that can make the degree of
    (f - bxkg) less than n.

    In other words the coefficient of xn in the polynomial (f - bxkg) has to be what?
     
  4. Mar 24, 2009 #3
    Re: polynomials

    the coefficient has to be 0 than I think? but degree k+m still could be greater than n as we dont know anything about k.
     
  5. Mar 25, 2009 #4

    epenguin

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    Re: polynomials

    You are just asked can you find, will you always be able to find, a k (and a b) that gives you the result you want?

    Maybe you would find it easier if you first considered a concrete case. I can choose any polynomials that come into my head as long as n >or= m.

    f = 5x3 + 10x2 + 2x + 8.5 and g = 3x + 2 comes into my head.

    Can you find a b and k for that that gives you a reduced degree result for (f - bxkg) ?
     
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