Examples of polynomials of single variable $s$ and polynomials in $N$ variables $s_1, s_2, cdots, s_n$ with...
$begingroup$
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
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
$endgroup$
add a comment |
$begingroup$
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
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
$endgroup$
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
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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
add a comment |
$begingroup$
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
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
$endgroup$
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
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
edited Dec 21 '18 at 18:48
DisintegratingByParts
59.2k42580
59.2k42580
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
asked Dec 21 '18 at 18:00
whydoesmathhatemewhydoesmathhateme
33
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
add a comment |
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
add a comment |
1 Answer
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active
oldest
votes
$begingroup$
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 $0=Q(x,y)=xy-a$
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
$endgroup$
add a comment |
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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 $0=Q(x,y)=xy-a$
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
$endgroup$
add a comment |
$begingroup$
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 $0=Q(x,y)=xy-a$
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
$endgroup$
add a comment |
$begingroup$
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 $0=Q(x,y)=xy-a$
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
$endgroup$
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 $0=Q(x,y)=xy-a$
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
edited Dec 21 '18 at 18:32
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
answered Dec 21 '18 at 18:25
zwimzwim
11.9k730
11.9k730
add a comment |
add a comment |
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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