Examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with...












0












$begingroup$


What are some examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients?



I have concocted the following examples for each. Do these seem right?



Polynomials of a single variable $s$:
begin{equation}
f(s) = a_0 + a_1s + a_2s^2
end{equation}



Polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients:
begin{equation}
f(s_1, s_2, cdots, s_n) = a_1s_1 + a_2s_2^2 + a_3 s_3^3 + cdots + a_n s_n^n
end{equation}






algebra-precalculus







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$endgroup$








  • 1




    $begingroup$
    You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
    $endgroup$
    – Caroline
    Dec 21 '18 at 18:03










  • $begingroup$
    Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
    $endgroup$
    – whydoesmathhateme
    Dec 21 '18 at 18:47










  • $begingroup$
    Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
    $endgroup$
    – John Douma
    Dec 21 '18 at 19:00
















0












$begingroup$


What are some examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients?



I have concocted the following examples for each. Do these seem right?



Polynomials of a single variable $s$:
begin{equation}
f(s) = a_0 + a_1s + a_2s^2
end{equation}



Polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients:
begin{equation}
f(s_1, s_2, cdots, s_n) = a_1s_1 + a_2s_2^2 + a_3 s_3^3 + cdots + a_n s_n^n
end{equation}






algebra-precalculus







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
    $endgroup$
    – Caroline
    Dec 21 '18 at 18:03










  • $begingroup$
    Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
    $endgroup$
    – whydoesmathhateme
    Dec 21 '18 at 18:47










  • $begingroup$
    Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
    $endgroup$
    – John Douma
    Dec 21 '18 at 19:00














0












0








0





$begingroup$


What are some examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients?



I have concocted the following examples for each. Do these seem right?



Polynomials of a single variable $s$:
begin{equation}
f(s) = a_0 + a_1s + a_2s^2
end{equation}



Polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients:
begin{equation}
f(s_1, s_2, cdots, s_n) = a_1s_1 + a_2s_2^2 + a_3 s_3^3 + cdots + a_n s_n^n
end{equation}






algebra-precalculus







share|cite|improve this question











$endgroup$




What are some examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients?



I have concocted the following examples for each. Do these seem right?



Polynomials of a single variable $s$:
begin{equation}
f(s) = a_0 + a_1s + a_2s^2
end{equation}



Polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with real coefficients:
begin{equation}
f(s_1, s_2, cdots, s_n) = a_1s_1 + a_2s_2^2 + a_3 s_3^3 + cdots + a_n s_n^n
end{equation}






algebra-precalculus




algebra-precalculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 21 '18 at 18:48









DisintegratingByParts

60k42681




60k42681










asked Dec 21 '18 at 18:00









whydoesmathhatemewhydoesmathhateme

33




33








  • 1




    $begingroup$
    You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
    $endgroup$
    – Caroline
    Dec 21 '18 at 18:03










  • $begingroup$
    Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
    $endgroup$
    – whydoesmathhateme
    Dec 21 '18 at 18:47










  • $begingroup$
    Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
    $endgroup$
    – John Douma
    Dec 21 '18 at 19:00














  • 1




    $begingroup$
    You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
    $endgroup$
    – Caroline
    Dec 21 '18 at 18:03










  • $begingroup$
    Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
    $endgroup$
    – whydoesmathhateme
    Dec 21 '18 at 18:47










  • $begingroup$
    Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
    $endgroup$
    – John Douma
    Dec 21 '18 at 19:00








1




1




$begingroup$
You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
$endgroup$
– Caroline
Dec 21 '18 at 18:03




$begingroup$
You are right - these are polynomials. I assume you user name has a reason? Right in the first lines of wikipedia you find some examples. en.wikipedia.org/wiki/Polynomial
$endgroup$
– Caroline
Dec 21 '18 at 18:03












$begingroup$
Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
$endgroup$
– whydoesmathhateme
Dec 21 '18 at 18:47




$begingroup$
Yes, you're right. It does has a reason. I'm currently self-studying functional analysis and at this point, I think math "hates" me. :) I hope to understand this topic a bit better in the near future.
$endgroup$
– whydoesmathhateme
Dec 21 '18 at 18:47












$begingroup$
Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
$endgroup$
– John Douma
Dec 21 '18 at 19:00




$begingroup$
Studying functional analysis before knowing what a polynomial is may be the source of your trouble.
$endgroup$
– John Douma
Dec 21 '18 at 19:00










1 Answer
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oldest

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$begingroup$

You are too restrictive for your example of a polynomial with multiple variables. The example you gave is indeed a polynomial but not all have this form.



In fact you can have any product of $s_i$, one part could be for instance $7s_1s{_3}^2{s_4}{s_5}^3$ and a polynomial would be a sum of these terms.



A polynomial of degree $3$ in $x,y$ have up to $10$ coefficients (of course some of them could be zero):
$$P(x)=(a_0)+(a_1x+b_1y)+(a_2x^2+b_2xy+c_2y^2)+(a_3x^3+b_3x^2y+c_3xy^2+d_3y^3)$$



I grouped terms that have the same overall degree:



$$deg(x^alpha y^beta z^gammacdots)=alpha+beta+gamma+cdots$$




  • the general form of a polynomial in $1$ variable is $displaystyle P(s)=sumlimits_{i=0}^{n} a_is^i$


  • the general form of a polynomial in $m$ variables is $displaystyle P(s_1,s_2,cdots,s_m)=sumlimits_{i=0}^{n}sumlimits_{|alpha|=i} a(alpha) {s_1}^{alpha_1}{s_2}^{alpha_2}cdots{s_m}^{alpha_m}$



with $alpha=(alpha_1,alpha_2,cdots,alpha_m)inmathbb N^m$ and $|alpha|=sumlimits_{j=1}^m alpha_j$




A simple example would be for instance the equation of a circle of centre $(a,b)$ and radius $r$.



The equation is $P(x)=0$ where $P$ is a polynomial $P(x,y)=(x-a)^2+(y-b)^2-r^2$



An hyperbola also has a polynomial equation $0=Q(x,y)=xy-a$






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    $begingroup$

    You are too restrictive for your example of a polynomial with multiple variables. The example you gave is indeed a polynomial but not all have this form.



    In fact you can have any product of $s_i$, one part could be for instance $7s_1s{_3}^2{s_4}{s_5}^3$ and a polynomial would be a sum of these terms.



    A polynomial of degree $3$ in $x,y$ have up to $10$ coefficients (of course some of them could be zero):
    $$P(x)=(a_0)+(a_1x+b_1y)+(a_2x^2+b_2xy+c_2y^2)+(a_3x^3+b_3x^2y+c_3xy^2+d_3y^3)$$



    I grouped terms that have the same overall degree:



    $$deg(x^alpha y^beta z^gammacdots)=alpha+beta+gamma+cdots$$




    • the general form of a polynomial in $1$ variable is $displaystyle P(s)=sumlimits_{i=0}^{n} a_is^i$


    • the general form of a polynomial in $m$ variables is $displaystyle P(s_1,s_2,cdots,s_m)=sumlimits_{i=0}^{n}sumlimits_{|alpha|=i} a(alpha) {s_1}^{alpha_1}{s_2}^{alpha_2}cdots{s_m}^{alpha_m}$



    with $alpha=(alpha_1,alpha_2,cdots,alpha_m)inmathbb N^m$ and $|alpha|=sumlimits_{j=1}^m alpha_j$




    A simple example would be for instance the equation of a circle of centre $(a,b)$ and radius $r$.



    The equation is $P(x)=0$ where $P$ is a polynomial $P(x,y)=(x-a)^2+(y-b)^2-r^2$



    An hyperbola also has a polynomial equation $0=Q(x,y)=xy-a$






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      You are too restrictive for your example of a polynomial with multiple variables. The example you gave is indeed a polynomial but not all have this form.



      In fact you can have any product of $s_i$, one part could be for instance $7s_1s{_3}^2{s_4}{s_5}^3$ and a polynomial would be a sum of these terms.



      A polynomial of degree $3$ in $x,y$ have up to $10$ coefficients (of course some of them could be zero):
      $$P(x)=(a_0)+(a_1x+b_1y)+(a_2x^2+b_2xy+c_2y^2)+(a_3x^3+b_3x^2y+c_3xy^2+d_3y^3)$$



      I grouped terms that have the same overall degree:



      $$deg(x^alpha y^beta z^gammacdots)=alpha+beta+gamma+cdots$$




      • the general form of a polynomial in $1$ variable is $displaystyle P(s)=sumlimits_{i=0}^{n} a_is^i$


      • the general form of a polynomial in $m$ variables is $displaystyle P(s_1,s_2,cdots,s_m)=sumlimits_{i=0}^{n}sumlimits_{|alpha|=i} a(alpha) {s_1}^{alpha_1}{s_2}^{alpha_2}cdots{s_m}^{alpha_m}$



      with $alpha=(alpha_1,alpha_2,cdots,alpha_m)inmathbb N^m$ and $|alpha|=sumlimits_{j=1}^m alpha_j$




      A simple example would be for instance the equation of a circle of centre $(a,b)$ and radius $r$.



      The equation is $P(x)=0$ where $P$ is a polynomial $P(x,y)=(x-a)^2+(y-b)^2-r^2$



      An hyperbola also has a polynomial equation $0=Q(x,y)=xy-a$






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        You are too restrictive for your example of a polynomial with multiple variables. The example you gave is indeed a polynomial but not all have this form.



        In fact you can have any product of $s_i$, one part could be for instance $7s_1s{_3}^2{s_4}{s_5}^3$ and a polynomial would be a sum of these terms.



        A polynomial of degree $3$ in $x,y$ have up to $10$ coefficients (of course some of them could be zero):
        $$P(x)=(a_0)+(a_1x+b_1y)+(a_2x^2+b_2xy+c_2y^2)+(a_3x^3+b_3x^2y+c_3xy^2+d_3y^3)$$



        I grouped terms that have the same overall degree:



        $$deg(x^alpha y^beta z^gammacdots)=alpha+beta+gamma+cdots$$




        • the general form of a polynomial in $1$ variable is $displaystyle P(s)=sumlimits_{i=0}^{n} a_is^i$


        • the general form of a polynomial in $m$ variables is $displaystyle P(s_1,s_2,cdots,s_m)=sumlimits_{i=0}^{n}sumlimits_{|alpha|=i} a(alpha) {s_1}^{alpha_1}{s_2}^{alpha_2}cdots{s_m}^{alpha_m}$



        with $alpha=(alpha_1,alpha_2,cdots,alpha_m)inmathbb N^m$ and $|alpha|=sumlimits_{j=1}^m alpha_j$




        A simple example would be for instance the equation of a circle of centre $(a,b)$ and radius $r$.



        The equation is $P(x)=0$ where $P$ is a polynomial $P(x,y)=(x-a)^2+(y-b)^2-r^2$



        An hyperbola also has a polynomial equation $0=Q(x,y)=xy-a$






        share|cite|improve this answer











        $endgroup$



        You are too restrictive for your example of a polynomial with multiple variables. The example you gave is indeed a polynomial but not all have this form.



        In fact you can have any product of $s_i$, one part could be for instance $7s_1s{_3}^2{s_4}{s_5}^3$ and a polynomial would be a sum of these terms.



        A polynomial of degree $3$ in $x,y$ have up to $10$ coefficients (of course some of them could be zero):
        $$P(x)=(a_0)+(a_1x+b_1y)+(a_2x^2+b_2xy+c_2y^2)+(a_3x^3+b_3x^2y+c_3xy^2+d_3y^3)$$



        I grouped terms that have the same overall degree:



        $$deg(x^alpha y^beta z^gammacdots)=alpha+beta+gamma+cdots$$




        • the general form of a polynomial in $1$ variable is $displaystyle P(s)=sumlimits_{i=0}^{n} a_is^i$


        • the general form of a polynomial in $m$ variables is $displaystyle P(s_1,s_2,cdots,s_m)=sumlimits_{i=0}^{n}sumlimits_{|alpha|=i} a(alpha) {s_1}^{alpha_1}{s_2}^{alpha_2}cdots{s_m}^{alpha_m}$



        with $alpha=(alpha_1,alpha_2,cdots,alpha_m)inmathbb N^m$ and $|alpha|=sumlimits_{j=1}^m alpha_j$




        A simple example would be for instance the equation of a circle of centre $(a,b)$ and radius $r$.



        The equation is $P(x)=0$ where $P$ is a polynomial $P(x,y)=(x-a)^2+(y-b)^2-r^2$



        An hyperbola also has a polynomial equation $0=Q(x,y)=xy-a$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 21 '18 at 18:32

























        answered Dec 21 '18 at 18:25









        zwimzwim

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