B About a definition: What is the number of terms of a polynomial P(x)?

[littlemathquark]

How many terms are in the expression:
$$\cos^2x - \sin^2x$$Or, should that be:
$$0+ 0\cos x +0\sin x + \cos^2x -\sin^2x$$Or, it could be any number greater than that.

But this example not polynomial.

 

[weirdoguy]

It is "polynomial in sine and cosine" as some would say. Anyways, it doesn't matter, the problem is the same.

 

[mathwonk]

As I tried, unsuccessfully it seems, to point out, you must first define what you mean by the coefficients of a polynomial. I.e. you must decide whether or not to include coefficients which are zero, among the coefficients.

In the case of 3X^5 +X^2 -4, you must decide what are the coefficients of this polynomial. The answer depends on your idea of what a polynomial is. If you think, as would be quite understandable, that it is what it looks like, i.e. (either 0 or) a finite sum of non-zero multiples of different powers of X, then you might well consider this one to have only three coefficients, (-4,1,3).

Other people might want to include the zero coefficients of the lower powers X, X^3, X^4, and say the coefficients are, in increasing order, (-4,0,1,0,0,3). I myself, using the mathematician's definition of the polynomial ring as an infinite graded direct sum, would say the (infinite) sequence of coefficients is (-4,0,1,0,0,3,0,0,........)

I.e. to answer questions about the coefficients of a polynomial, you must first define what the coefficients of a polynomial are. I can think of three different ways to define "the coefficients" of your polynomial.

1. the coefficients are the non-zero coefficients, i.e. a_0 = -4, a_2 = 1, a_5 = 3, and all others are undefined.

2. the coefficients are those of degree ≤ 5, i.e. a_0 = -4, a_1 = 0, a_2 = 1, a_3 = a_4 = 0, a_5 = 3, and all others are undefined.

3. the coefficients are those of all degrees, i.e.
a_0 = -4, a_1 = 0, a_2 = 1, a_3 = a_4 = 0, a_5 = 3, a_j = 0 for j ≥ 6. This is actually quite common, i.e. most people would probably say the coefficient of X^7 in your polynomial is zero.

A fourth way is to define the coefficients, not of a polynomial, but of a particular finite representation of a polynomial. I.e. if you consider a polynomial to be, as it is often defined, a finite linear combination
a_0 +a_1X + a_2X^2 +......+a_nX^n, of powers of X, then you might want to say the coefficients of this "polynomial" are the n+1 elements of the sequence a_0,........,a_n.

Unfortunately, since some of these may be zero, in particular a_n may be zero, this sequence changes with different representations of the same polynomial. If you say further that a_n ≠ 0, you get our definition #2 above, but it is unusual for a book to define only such non-zero polynomials.

I.e. with this definition of "polynomial" you must say, as some books forget to do, [Dummit-Foote, Abstract Algebra, 3rd ed. p.234], but some do not forget [Van der Waerden, Modern Algebra, trans. by Blum from 2nd German edition, pp.45-6], that two different such finite linear combinations define the same polynomial if and only if their non-zero coefficients are all the same. This of course destroys the uniqueness of this definition of the coefficients, since with this definition, different representations of the same polynomial have different coefficients.

So in order to answer your questions, first make up your mind what you mean by "the coefficients" of your polynomial, and again I recommend reading Mike Artin's discussion of this situation, on pp.350-351, of his Algebra. Or else, just say "non-zero coefficients", if that is what you want in your questions, as others have suggested.

 

[mathwonk]

This confusion arises already in vector space theory. I.e. if e1 = (1,0,0), e2 = (0,1,0), and e3 = (0,0,1), is the standard basis of R^3, then what are the "coefficients" of the vector v = 3.e1 + 7.e2 = (3,7,0)? are they the coefficients 3 and 7 used in the given linear combination, or are they the coordinates (3,7,0) of the final result?

To be clear you have to say what you mean, i.e. you must distinguish between the coefficients used in a particular linear combination, and those that could be used in some standard universal linear combination for the same vector.

In this setting, the two distinct words "coefficient" and "coordinate" can be used to offer some additional clarity. I.e. we usually use "coefficient" in referring to those occurring in a particular linear combination, and "coordinate" when referring to the coefficients of the universal linear combination involving the whole basis.

We could do this for polynomials as well, i.e. refer to the "coefficients" of a particular polynomial representation, and the "coordinates" in the full infinite sequence of them. But this will never catch on after all these years. So you will still always have to ask someone what they mean by the phrase "coefficients of a polynomial".

More agreement may be found if you ask instead for the "nth coefficient". I.e. oddly enough, some people may say that "the coefficients" of -4 +X^2 + 3X^5 are -4, 1, and 3, but they may also say that the "6th coefficient", i.e. a_6, is zero.

I just noticed that actually this polynomial does not have a visible quadratic coefficient, and that its absence is assumed to mean that a_2 = 1. I.e. this "polynomial" is not actually a finite sum of terms like a_jX^j. In other places, like the cubic coefficient, its absence is assumed to mean a_3 = 0.

I note that stating this convention is also forgotten in some books, e.g. even Van de Waerden, which does say that the omission of the term a_jX^j means that a_j = 0, does not say that the omission of just a_j means that a_j = 1.

 

[mathwonk]

Even simpler, how many "digits" does the number
101 = 1.(10)^0 + 1.(10)^2
= 1(10)^0 + 0.(10)^1 + 1.(10)^2
= 1.(10)^0 + 0.(10)^1 + 1.(10)^2 + 0.(10)^3. have ?

Of course here there is general agreement here, which aligns with the "degree (+1)" for polynomials,(as required by positional notation, which omits the actual basis vectors).

But on the second line of a personal check e.g., we return to the standard "polynomial" expression, writing "one hundred one", omitting the tens coefficient.

(I had forgotten that positional notation just means viewing the non - negative integers notationally as an infinite graded direct sum of copies of the group Z/(10), but with different operations, incorporating "carrying". E.g. when adding tens, you go around in a circle with ten spaces, but every time you pass "go" you collect $100.)

 

[littlemathquark]

I encountered the following question on the topic and came up with two solutions. I couldn't decide which one is correct:
Let $a$ and $b$ be two real numbers. One of the terms of the $7-term$ polynomial $P(x)=x(x-1)^a$ is $b.x^5$. Find sum of $a+b$
First Solution: there are $n+1$ terms in the expansion of $(x+y)^n$ so $a=6$ Then
$p(x)=x(x-1)^6$ so $bx^5=15x^5$ and $b=15$
$a+b=6+15=21$

Second solution: İf $p(x)$ has $7$ terms polynomial then it must be $6.$ degree; that is $a=5$ so $p(x)=x(x-1)^5$ and $bx^5=-5x^5$ and $b=-5$
$a+b=5+-5=0$

 

[martinbn]

I encountered the following question on the topic and came up with two solutions. I couldn't decide which one is correct:
Let $a$ and $b$ be two real numbers. One of the terms of the $7-term$ polynomial $P(x)=x(x-1)^a$ is $b.x^5$. Find sum of $a+b$
First Solution: there are $n+1$ terms in the expansion of $(x+y)^n$ so $a=6$ Then
$p(x)=x(x-1)^6$ so $bx^5=15x^5$ and $b=5$
$a+b=6+5=11$

Second solution: İf $p(x)$ has $7$ terms polynomial then it must be $6.$ degree; that is $a=5$ so $p(x)=x(x-1)^5$ and $bx^5=-5x^5$ and $b=-5$
$a+b=5+-5=0$

As I have been saying there are no absolute definitions. It depends on the context, and what definition your source uses. Where did you encounter this? Earlier you said that you are making this questions up, now that you encountered them somewhere. The more information you give the easier the discussion will be.

 

[littlemathquark]

A friend of mine asked the question and there is no information about context and definitions.

 

[martinbn]

A friend of mine asked the question and there is no information about context and definitions.

Then there is no answer to your question which solution is correct. Each of them is correct if you include the reason how you determined $a$.

 

[littlemathquark]

Ok, thank you.
İs these true ?
if $p(x)=x^7$ is an algebraic expression then it's s only one term but if $p(x)=x^7$ is an polynomial there may be 7+1=8 terms according to context.

 

[martinbn]

Ok, thank you.
İs these true ?
if $p(x)=x^7$ is an algebraic expression then it's s only one term but if $p(x)=x^7$ is an polynomial there may be 7+1=8 terms according to context.

There may be also 10 terms if you consider it as a polynomial in the space of polynomials of digree up to nine. It can have infinitely many if you think of polynomials as sequences.

 

[Mark44]

İs these true ?
if $p(x)=x^7$ is an algebraic expression then it's s only one term but if $p(x)=x^7$ is an polynomial there may be 7+1=8 terms according to context.

The $x^7$ part is an expression but $p(x) = x^7$ is an equation. As you said, the right side consists of one term.
Very few people would insist that $x^7$ is a polynomial with any more than one term, let alone eight of them.

 

[littlemathquark]

As I tried, unsuccessfully it seems, to point out, you must first define what you mean by the coefficients of a polynomial. I.e. you must decide whether or not to include coefficients which are zero, among the coefficients.

In the case of 3X^5 +X^2 -4, you must decide what are the coefficients of this polynomial. The answer depends on your idea of what a polynomial is. If you think, as would be quite understandable, that it is what it looks like, i.e. (either 0 or) a finite sum of non-zero multiples of different powers of X, then you might well consider this one to have only three coefficients, (-4,1,3).

Other people might want to include the zero coefficients of the lower powers X, X^3, X^4, and say the coefficients are, in increasing order, (-4,0,1,0,0,3). I myself, using the mathematician's definition of the polynomial ring as an infinite graded direct sum, would say the (infinite) sequence of coefficients is (-4,0,1,0,0,3,0,0,........)

I.e. to answer questions about the coefficients of a polynomial, you must first define what the coefficients of a polynomial are. I can think of three different ways to define "the coefficients" of your polynomial.

1. the coefficients are the non-zero coefficients, i.e. a_0 = -4, a_2 = 1, a_5 = 3, and all others are undefined.

2. the coefficients are those of degree ≤ 5, i.e. a_0 = -4, a_1 = 0, a_2 = 1, a_3 = a_4 = 0, a_5 = 3, and all others are undefined.

3. the coefficients are those of all degrees, i.e.
a_0 = -4, a_1 = 0, a_2 = 1, a_3 = a_4 = 0, a_5 = 3, a_j = 0 for j ≥ 6. This is actually quite common, i.e. most people would probably say the coefficient of X^7 in your polynomial is zero.

A fourth way is to define the coefficients, not of a polynomial, but of a particular finite representation of a polynomial. I.e. if you consider a polynomial to be, as it is often defined, a finite linear combination
a_0 +a_1X + a_2X^2 +......+a_nX^n, of powers of X, then you might want to say the coefficients of this "polynomial" are the n+1 elements of the sequence a_0,........,a_n.

Unfortunately, since some of these may be zero, in particular a_n may be zero, this sequence changes with different representations of the same polynomial. If you say further that a_n ≠ 0, you get our definition #2 above, but it is unusual for a book to define only such non-zero polynomials.

I.e. with this definition of "polynomial" you must say, as some books forget to do, [Dummit-Foote, Abstract Algebra, 3rd ed. p.234], but some do not forget [Van der Waerden, Modern Algebra, trans. by Blum from 2nd German edition, pp.45-6], that two different such finite linear combinations define the same polynomial if and only if their non-zero coefficients are all the same. This of course destroys the uniqueness of this definition of the coefficients, since with this definition, different representations of the same polynomial have different coefficients.

So in order to answer your questions, first make up your mind what you mean by "the coefficients" of your polynomial, and again I recommend reading Mike Artin's discussion of this situation, on pp.350-351, of his Algebra. Or else, just say "non-zero coefficients", if that is what you want in your questions, as others have suggested.

Let $p(x)=3x^4+x^2-4$ and $Q(x)=3x^4+(a-1)x^3+x^2+(b+2)x-4$ if $p(x)=Q(x)$ then find $a+b$
In the first method of defining coefficients, how can the problem above be solved when the coefficients of $x^3$ and $x$ in the polynomial $p$ are undefined? There's no problem if we assume these coefficients are $0$, but what do we do when they are undefined?"

 

[weirdoguy]

what do we do when they are undefined

We realise that this definition is not appropriate for this problem, and then we adapt definition 2, just like the person who came with this exercise did. And then we don't waste any more time on such nonimportant issues.

 

[martinbn]

Let $p(x)=3x^4+x^2-4$ and $Q(x)=3x^4+(a-1)x^3+x^2+(b+2)x-4$ if $p(x)=Q(x)$ then find $a+b$
In the first method of defining coefficients, how can the problem above be solved when the coefficients of $x^3$ and $x$ in the polynomial $p$ are undefined? There's no problem if we assume these coefficients are $0$, but what do we do when they are undefined?"

What if $Q(x)=cx^5+\cdots$? Does $p(x)$ have a term of degree five or not?

 

[littlemathquark]

What if $Q(x)=cx^5+\cdots$? Does $p(x)$ have a term of degree five or not?

According to first way of definition of coefficient, $x^5$ term is undefined.

 

[MidgetDwarf]

Or is of degree zero -- probably a better way to say the same thing.

The polynomial of degree 0 are those for which the polynomial is constant.

The zero polynomial has degree - infinity.

 

[jbriggs444]

The zero polynomial has degree - infinity.

That claim is a bit too strong. Wikipedia defines the degree of a polynomial as:

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.

If we take that definition at face value, the degree of the zero polynomial would be the same as the maximum value in the empty set. Which should (in my opinion) be left undefined. Though one can find other opinions.

Like all definitions, this is a matter of convenience and convention rather than a matter of underlying mathematical truth. The Wikipedia article goes on to point out that conventions vary:

The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or −∞)

If we look at the footnote for this claim, we find:

Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)
Childs (1995) uses −1. (p. 233)
Childs (2009) uses −∞ (p. 287), however he excludes zero polynomials in his Proposition 1 (p. 288) and then explains that the proposition holds for zero polynomials "with the reasonable assumption that −∞ + m = −∞ for m any integer or m = −∞".
Axler (1997) uses −∞. (p. 64)
Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as −1 ∈ Z or as −∞, as long as deg 0 < deg A for all A ≠ 0." (A is a polynomial.) However, he excludes zero polynomials in his Proposition 5.3. (p. 121)

 

[A.T.]

A friend of mine asked the question and there is no information about context and definitions.

Then ask your 'friend' for the context.