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Condition for this polynomial to be a perfect square

  1. Oct 27, 2012 #1

    utkarshakash

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    Gold Member

    1. The problem statement, all variables and given/known data
    The condition that [itex]x^4+ax^3+bx^2+cx+d [/itex] is a perfect square, is

    2. Relevant equations

    3. The attempt at a solution
    If the above polynomial will be a perfect square then it can be represented as

    [itex](x-\alpha)^2(x-\beta)^2[/itex] where α and β are the roots of it.This means that two roots of it will be identical.
    Am I correct in my assumption?
     
  2. jcsd
  3. Oct 27, 2012 #2

    Mentallic

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    Homework Helper

    Why do the 2 roots have to be identical? You've already shown what the general form of a quartic (degree 4 polynomial) needs to be in for it to be a perfect square, so go ahead and expand.

    Although, I'd suggest you make the general form a general quadratic squared, so you'll have

    [tex](x^2+mx+n)^2[/tex]

    And once you expand this out, you can compare coefficients and find the conditions on a,b,c,d as required.
     
  4. Oct 27, 2012 #3
    What does it mean to be a perfect square?

    If the equation is quadratic ax^2+bx+c then it is a perfect square if it can be factored as (x+k)^2, ie two identical factors.

    If you're starting with a fourth degree polynomial and you want to constrain it to be a perfect square then it must be the product of two identical factors. That factor must be a polynomial of degree 2 since you need an x^4 term when it is multiplied out. The most general form of a second degree polynomial is given above.
     
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