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Polynomial question, am I correct?

  1. Nov 10, 2007 #1
    [SOLVED] Polynomial question, am I correct?

    1. The problem statement, all variables and given/known data
    p(z) = z[tex]^{5}[/tex]-5z[tex]^{4}[/tex]+18z[tex]^{3}[/tex]-50z[tex]^{2}[/tex]+81z-45
    Show that z - 1 is a factor of p(z)


    2. Relevant equations

    None

    3. The attempt at a solution

    Well since 1 is a root of p(z), then p(1) = 0
    Then z - 1 is a factor?
     
  2. jcsd
  3. Nov 10, 2007 #2

    PhY

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    p(1) is equal to 0, than yes it is a factor.
     
  4. Nov 10, 2007 #3
    I know it is a factor, but is my method correct for 'showing' that it is so?
     
  5. Nov 10, 2007 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    That was PhY's answer! (less a stray "e"). Because p(1) is equal to 0, then yes it is a factor.

    You are using a very simple fact about polynomials- If P(x) is any polynomial, dividing P(x) by (x-a) results in a quotient, Q(x), and a remainder, r, which, since x-a has degree 1, must be a number: P(x)= Q(x)(x-a)+ r. Now, let x= a: P(a)= Q(x)(a- a)+ r= r. If P(a)= 0 then r must be 0 and so (x- a) is a factor of P(x).
     
  6. Nov 10, 2007 #5
    Thanks for clearing that up.
     
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