Name: (1+x)^3 vs. 1+3*x^1+3*x^2+x^3
Prof. Oro Yusuf
(1+x)^3 = 1+3*x^1+3*x^2+x^3
In the above equation, what are the expressions "(1+x)^3" and
"1+3*x^1+3*x^2+x^3" called? I know it involves factoring and
distribution and "1+3*x^1+3*x^2+x^3" is both a polynomial and
the binomial series expansion of (1+x)^3, but are there more
established and distinct terms for "(1+x)^n" and its companion
"1+...+x^n"?
As long as n is a positive integer, then both are closed-form
expressions, so that doesn't help define the difference between
the two forms.
William Elliot
> A quikkie basic algebra question:
>
> (1+x)^3 = 1+3*x^1+3*x^2+x^3
>
> In the above equation, what are the expressions "(1+x)^3" and
The cube of a linear expression.
> "1+3*x^1+3*x^2+x^3" called?
An unreadably dense series of symbols.
1 + 3x + 3x^2 + x^3
is a third degree polynomial with four terms.
> I know it involves factoring and
> distribution and "1+3*x^1+3*x^2+x^3" is both a polynomial and
> the binomial series expansion of (1+x)^3,
The factors of the polynomial
1 + 3x + 3x^2 + x^3
are three 1 + x, ie (1 + x)^3.
> but are there more established and distinct terms for "(1+x)^n" and its
> companion "1+...+x^n"?
Is the companion sum(j=0,n) x^n?
> As long as n is a positive integer, then both are closed-form
> expressions, so that doesn't help define the difference between
> the two forms.
>
(1 + x)^3 is a multiplication of factors.
1 + 3x + 3x^2 + x^3 is a sum of terms.
1 + 3x + 3x^2 + x^3 is a simplification of (1 + x)^3.
(1 + x)^3 is a factorization of 1 + 3x + 3x^2 + x^3.
Gottfried Helms
same function of x, (1+x)^3 could be called "the monomial form"
and 1 + 3x + 3x^2 + x^3 "the polynomial form" of the function.
However - this is just an ad-hoc-invention, so...
Gottfried Helms
Prof. Oro Yusuf
On Apr 22, 3:58 am, William Elliot wrote:
> On Thu, 22 Apr 2010, Prof. Oro Yusuf wrote:
>> A quikkie basic algebra question:
>
>> (1+x)^3 = 1+3*x^1+3*x^2+x^3
>
>> In the above equation, what are the expressions "(1+x)^3" and
>
> The cube of a linear expression.
>
>> "1+3*x^1+3*x^2+x^3" called?
>
> An unreadably dense series of symbols.
>
> 1 + 3x + 3x^2 + x^3
>
> is a third degree polynomial with four terms.
Isn't "1" really "x^0", so shouldn't it really be a fourth degree
polynomial?
>> I know it involves factoring and
>> distribution and "1+3*x^1+3*x^2+x^3" is both a polynomial and
>> the binomial series expansion of (1+x)^3,
>
> The factors of the polynomial
> 1 + 3x + 3x^2 + x^3
>
> are three 1 + x, ie (1 + x)^3.
>
>> but are there more established and distinct terms for "(1+x)^n"
>> and its companion "1+...+x^n"?
>
> Is the companion sum(j=0,n) x^n?
No, sorry, by "established and distinct terms" I meant names, not
"formula terms"! :(
>> As long as n is a positive integer, then both are closed-form
>> expressions, so that doesn't help define the difference between
>> the two forms.
>
> (1 + x)^3 is a multiplication of factors.
> 1 + 3x + 3x^2 + x^3 is a sum of terms.
>
> 1 + 3x + 3x^2 + x^3 is a simplification of (1 + x)^3.
>
> (1 + x)^3 is a factorization of 1 + 3x + 3x^2 + x^3.
Okay here, but as I say above, what if n is negative or a fraction:
sqrt(1 + x) = 1 - 1/2x - 1/8x^2 - 1/16x^3 - ...C(1/2,oo)x^oo ?
Is "1 - 1/2x - 1/8x^2 - 1/16x^3 - ...C(1/2,oo)x^oo" an infinity degree
polynomial with infinity plus one terms?
If it does have a completely different name (I know it is sometimes
referred to as a "power series"), then what would the common name for
"1 + 3x + 3x^2 + x^3" and "1 - 1/2x - 1/8x^2 - 1/16x^3 -
...C(1/2,oo)x^oo" vs. "(1 + x)^3" and "sqrt(1 + x)" be?
Or for that matter, *any* "(a + b)^n", "(a + b + c)^n", etc., closed
forms vs. their series expansions?
Zdislav V. Kovarik
On Thu, 22 Apr 2010, Prof. Oro Yusuf wrote:
(1+x)^3
factored form
1+3*x^1+3*x^2+x^3
expanded form
Illustrated by MATLAB commands ("ans" means "answer"):
EDU syms x
EDU (1+x)^3
ans =
(1+x)^3
EDU expand(ans)
ans =
1+3*x+3*x^2+x^3
EDU factor(ans)
ans =
(1+x)^3
Cheers, ZVK(Slavek)
Kaimbridge
> References: <4BCFAD97...@m2n.org.uk>
>
> On Apr 22, 3:58 am, William Elliot wrote:
>> On Thu, 22 Apr 2010, Prof. Oro Yusuf wrote:
>>> A quikkie basic algebra question:
>>
>>> (1+x)^3 = 1+3*x^1+3*x^2+x^3
>>
>>> In the above equation, what are the expressions "(1+x)^3" and
>>
>> The cube of a linear expression.
>>
>>> "1+3*x^1+3*x^2+x^3" called?
>> 1 + 3x + 3x^2 + x^3
>>
>> is a third degree polynomial with four terms.
>>> I know it involves factoring and
>>> distribution and "1+3*x^1+3*x^2+x^3" is both a polynomial and
>>> the binomial series expansion of (1+x)^3,
>>> As long as n is a positive integer, then both are closed-form
>>> expressions, so that doesn't help define the difference between
>>> the two forms.
>>
>> (1 + x)^3 is a multiplication of factors.
>> 1 + 3x + 3x^2 + x^3 is a sum of terms.
>>
>> 1 + 3x + 3x^2 + x^3 is a simplification of (1 + x)^3.
>>
>> (1 + x)^3 is a factorization of 1 + 3x + 3x^2 + x^3.
>
> Okay here, but as I say above, what if n is negative or a fraction:
>
> sqrt(1 + x) = 1 - 1/2x - 1/8x^2 - 1/16x^3 - ...C(1/2,oo)x^oo ?
>
> Is "1 - 1/2x - 1/8x^2 - 1/16x^3 - ...C(1/2,oo)x^oo" an infinity
> degree polynomial with infinity plus one terms?
> If it does have a completely different name (I know it is
> sometimes referred to as a "power series"), then what would
> the common name for
> "1 + 3x + 3x^2 + x^3" and "1 - 1/2x - 1/8x^2 - 1/16x^3 -
> ...C(1/2,oo)x^oo" vs. "(1 + x)^3" and "sqrt(1 + x)" be?
> Or for that matter, *any* "(a + b)^n", "(a + b + c)^n", etc., closed
> forms vs. their series expansions?
For the equal factors that you have been talking about, I would say
"(1 + x)^n" is the binomial (or more generally, polynomial) raised
factor ("common binomial/polynomial factor"?), while "1 + C(n,1)x +
C(n,2)x^2 + C(n,3)x^3 + ...C(n,oo)x^oo" would be its distributed
binomial/polynomial series.
There is a whole "...For Dummies" section on distribution:
http://www.dummies.com/how-to/education-languages/math/algebra/distribution.html
An interesting aside to this is the fact that the basic elliptic
integral is nothing more than a modified distributed binomial
_
series, where "sin(x)^n" is replaced with "/ sin(x)^n dx" (with
_/
the cosine multiples form of "sin(x)^n" likewise adjusted). See:
https://kb.osu.edu/dspace/bitstream/1811/24409/1/Rapp_Geom_Geod_%20Vol_I.pdf
pp.9-11, then
https://kb.osu.edu/dspace/bitstream/1811/24409/1/Rapp_Geom_Geod_%20Vol_II.pdf
pp.7-10
Thus,
_L_f
/
/ (1- (sin(L)sin(oe))^2)^.5 dL
_/L_s
1 _L_f
= 1 - - / sin(L)^2dL sin(oe)^2
2 _/L_s
1 _L_f
- - / sin(L)^4dL sin(oe)^4
8 _/L_s
1 _L_f
- -- / sin(L)^6dL sin(oe)^6 - ...
16 _/L_s
Therefore, the elliptic integrand is a regular raised binomial
factor, "(1- (sin(L)sin(oe))^2)^.5", while the integral's
primitive is a modified distributed binomial series: If the
raised factor has a finite distributed series, then an elliptic
integral that has a raised factor integrand with a regular
finite distributed series, will have a primitive that is a
finite (albeit, modified) distributed series, i.e., what is
commonly referred to as a "closed-form solution"!
~Kaimbridge~
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Kaimbridge
> the cosine multiples form of "sin(x)^n" likewise adjusted). See:
>
> https://kb.osu.edu/dspace/bitstream/1811/24409/1/Rapp_Geom_Geod_%20Vo...
Make that
https://kb.osu.edu/dspace/bitstream/1811/24333/1/Rapp_Geom_Geod_Vol_I.pdf
> pp.9-11, then
>
> https://kb.osu.edu/dspace/bitstream/1811/24409/1/Rapp_Geom_Geod_Vol_II.pdf
> pp.7-10
Michael Stemper
>References: <4BCFAD97...@m2n.org.uk>
>On Apr 22, 3:58 am, William Elliot wrote:
> > On Thu, 22 Apr 2010, Prof. Oro Yusuf wrote:
> >> "1+3*x^1+3*x^2+x^3" called?
> >
> > An unreadably dense series of symbols.
> >
> > 1 + 3x + 3x^2 + x^3
> >
> > is a third degree polynomial with four terms.
>
>Isn't "1" really "x^0",
Yes, it is.
> so shouldn't it really be a fourth degree
>polynomial?
No, it shouldn't. The degree of a polynomial is *not* the number of
terms. The degree of a polynomial is the largest power of any term with
a non-zero coefficient. In this example, that term is x^3, which has
a coefficient of 1. Therefore, it's a third-degree polynomial, as was
correctly stated.
--
Michael F. Stemper
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