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Math Help - Roots and Solutions

  1. #1
    Senior Member pankaj's Avatar
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    Roots and Solutions

    What is difference between number of roots of a polynomial equation and number of solutions to a polynomial equation?

    Do they mean same thing OR can an equation have different number of roots and different number of solutions?
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  2. #2
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    They are the same...

    Unless they specify, in terms of solutions, that they need "real solutions," or "nontrivial solutions," etc. The roots are always the roots, but what is or is not an acceptable "solution" may depend upon how the question is asked.

    EDIT: Also to consider: The above discussion presupposes that by "polynomial equation" you mean one that is given in the format  a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0."
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  3. #3
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    In my precalc book, if you have a polynomial function f(x), the following are equivalent:

    * a root of the polynomial f(x),
    * a solution to the polynomial equation f(x) = 0,
    * a zero of the polynomial f(x), and
    * an x-intercept of the polynomial f(x) (assuming that we're only talking about real roots).


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    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by pankaj View Post
    What is difference between number of roots of a polynomial equation and number of solutions to a polynomial equation?

    Do they mean same thing OR can an equation have different number of roots and different number of solutions?
    I agree with Aleph, because the differences between the two are subtle at best. A solution is finding the values in an equation that satisyfy the given condition, and finding roots generally implies finding the zeros (a type of solution) that satisfy the given conditions. I would maybe say that the word "solution" is a bit more broad than "root".

    Read the first few lines of this:


    Roots or zeros of a polynomial - Topics in precalculus
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    Quote Originally Posted by VonNemo19 View Post
    Okay, that was scary -- I was looking at the same exact site just minutes ago!


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    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by yeongil View Post
    Okay, that was scary -- I was looking at the same exact site just minutes ago!


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    Haha. That's google for you.
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    One other consideration comes to mind here. Sometimes roots are counted with multiplicity, and sometimes without. So for instance

    (x-2)^5=0

    has 5 roots counting multiplicity, and 1 root, not counting multiplicity. In either case, there's really only 1 "solution."
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  8. #8
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by AlephZero View Post
    One other consideration comes to mind here. Sometimes roots are counted with multiplicity, and sometimes without. So for instance

    (x-2)^5=0

    has 5 roots counting multiplicity, and 1 root, not counting multiplicity. In either case, there's really only 1 "solution."
    Good example, man!
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  9. #9
    Senior Member pankaj's Avatar
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    Quote Originally Posted by AlephZero View Post
    One other consideration comes to mind here. Sometimes roots are counted with multiplicity, and sometimes without. So for instance

    (x-2)^5=0

    has 5 roots counting multiplicity, and 1 root, not counting multiplicity. In either case, there's really only 1 "solution."
    Yes.This is the part I wanted to confirm.Could you provide more examples like this this.
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  10. #10
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    Quote Originally Posted by pankaj View Post
    Yes.This is the part I wanted to confirm.Could you provide more examples like this this.
    OK, well, any polynomial of the form a_nx^n + a_{n-1}x^{n-1}+\ldots+a_1x+a_0=0 can be factored in terms of its roots r_1, r_2, \ldots, r_k, with k\leq n, and corresponding multiplicities m_1, m_2, \ldots, m_k thusly:

    c_0(x-r_1)^{m_1}(x-r_2)^{m_2}\cdots(x-r_k)^{m_k}=0.

    Here c_0 is just a constant. There will be k distinct roots, and therefore k "solutions." But if we include multiplicity, there will be n roots.

    So in general, the amount of "solutions" is less than or equal to the amount of roots, depending on whether or not we count repeated roots.

    I hope that answers your question.
    Last edited by AlephZero; August 1st 2009 at 07:14 PM. Reason: clarity
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