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

[littlemathquark]

TL;DR Summary

Let polynomial $P(x)=3x^5+x^2-4$ so what is the number of terms of $P(x)$? İt's 5+1=6 terms because of $degP(x)=5$ (since $P(x)=3x^5+0.x^4+0.x^3+x^2-4$) or only three terms? Why?

Let polynomial $P(x)=3x^5+x^2-4$ so what is the number of term of $P(x)$? İt's 5+1=6 terms because of $degP(x)=5$ (since $P(x)=3x^5+0.x^4+0.x^3+x^2-4$) or only three terms? Why?

 

[mathisrad]

From my understanding, the definition of a "term" in a polynomial refers to the individual pieces (non-zero) separated by addition or subtraction notation. So in the above example, I would consider it having 3 terms. A monomial doesn't mean it has to be degree one; 5x^2 is also a monomial, for example, so by this logic, P(x) would be considered a trinomial. The only thing I can think of that relates a degree to the number of terms is the n+1 rule you stated above.

 

[FactChecker]

From my understanding, the definition of a "term" in a polynomial refers to the individual pieces (non-zero) separated by addition or subtraction notation.

That is how I use "term". I don't know if there is an official definition. If two things are added, I call it two terms no matter what the nature of those two things are. I would say that $5x^{10} +x^{10} +\sin(x)$ has three terms.
The only exception is if is a polynomial. In that case "the 5th term of the polynomial" might be $0=0\cdot x^4$.

 

[martinbn]

TL;DR Summary: Let polynomial $P(x)=3x^5+x^2-4$ so what is the number of term of $P(x)$? İt's 5+1=6 terms because of $degP(x)=5$ (since $P(x)=3x^5+0.x^4+0.x^3+x^2-4$) or only three terms? Why?

Let polynomial $P(x)=3x^5+x^2-4$ so what is the number of term of $P(x)$? İt's 5+1=6 terms because of $degP(x)=5$ (since $P(x)=3x^5+0.x^4+0.x^3+x^2-4$) or only three terms? Why?

If you allow zero, then why do you count only these terms? For example $0.x^{20}$ is also there.

 

[littlemathquark]

If you allow zero, then why do you count only these terms? For example $0.x^{20}$ is also there.

because DegP(x)=5

 

[littlemathquark]

Which is true?
$P(x) =3x^5+x^2-4$ has three terms ($3x^5,x^2 and - 4)$ or $P(x)$ has 6 terms ($3x^5,0.x^4,0.x^3,x^2,0.x and - 4$) and which must prefer, why?

 

[martinbn]

Which is true?
$P(x) =3x^5+x^2-4$ has three terms ($3x^5,x^2 and - 4)$ or $P(x)$ has 6 terms ($3x^5,0.x^4,0.x^3,x^2,0.x and - 4$) and which must prefer, why?

As with definitions there is no true or false it is a matter of convention. If you want to be clear you can say three non-zero terms. Also if it is the second possiblility and you add one more term, say $x^{100}$, how many terms do you have now?

 

[littlemathquark]

As with definitions there is no true or false it is a matter of convention. If you want to be clear you can say three non-zero terms. Also if it is the second possiblility and you add one more term, say $x^{100}$, how many terms do you have now?

101 terms.
But sometimes P(x) is called "trinom" but it's 6 terms.

 

[mathisrad]

The name trinomial refers to three non-zero terms(at least that's how I treat it), which is the key difference between the two definitions you offer. P(x) is a trinomial because, by definition, it has three non-zero terms. I don't think I've seen anyone refer to something based on its max number of terms.

 

[Mark44]

$P(x) =3x^5+x^2-4$ has three terms

Yes.

The name trinomial refers to three non-zero terms(at least that's how I treat it)

I agree. When we talk about the terms of a polynomial, we're referring to the ones that have a nonzero coefficient.

 

[Gavran]

0 is a term with no degree and it is not included into the number of terms of a polynomial. For example, the zero polynomial P(x)=0 has zero terms and it has no degree.

 

[PeroK]

Unless the number of terms excludes zero terms, then it's difficult to see how it can be well-defined. Especially more generally where there is no upper limit imposed by the degree of a polynomial.

 

[Mark44]

the zero polynomial P(x)=0 has zero terms and it has no degree.

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

 

[littlemathquark]

Unless the number of terms excludes zero terms, then it's difficult to see how it can be well-defined. Especially more generally where there is no upper limit imposed by the degree of a polynomial.

Could you elaborate a little more on what you mean and provide an example? Thank you.

 

[PeroK]

Could you elaborate a little more on what you mean and provide an example? Thank you.

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.

 

[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.

Thank you. Also İf for example $P(x)=0x^8+0x^7+0x^6+3x^5+x^2-4$ we can't say degree of $P(x)$. Am I right?

 

[PeroK]

Thank you. Also İf for example $P(x)=0x^8+0x^7+0x^6+3x^5+x^2-4$ we can't say degree of $P(x)$. Am I right?

I'm not really sure how we ended up discussing this.

 

[martinbn]

Another reason why it is not useful to define it as the degree plus one, is that it brings no new information, the degree is enough.

 

[SammyS]

0 is a term with no degree and it is not included into the number of terms of a polynomial. For example, the zero polynomial P(x)=0 has zero terms and it has no degree.

I agree.

The degree of the univariate polynomial function, $P(x) = 0 \,, \$ is usually said to to be undefined. In other words, it has no degree.

Depending upon context, you may alternatively see the zero polynomial be assigned the degree of $-1$ or perhaps $-\infty$ .

 

[mathwonk]

It is meaningless to discuss the correct use of a word in mathematics unless you first define that word. Here the undefined word is "term". A similar confusion arises in the use of the word "coefficient" for polynomials.

For instance you might say that the "nth term" of a polynomial P, is a_n.X^n, where a_n is the nth coefficient of P. Then of course you must define "nth coefficient". Thus an equally confusing question is to ask how many coefficients a polynomial P has. Since technically a polynomial is defined as an infinite sequence (a_0, a_1,.........., a_n,..........), with all but a finite number of entries equal to zero, there are always infinitely many coefficients, mostly zero. I.e. a polynomial over a ring A, is a mapping a from the non negative integers into A, with only a finite number of non-zero values, and the nth coefficient is a(n).

One may thus consider the "coefficients" of P to be either the full infinite sequence (a_0,....,a_n,..........), or (for a non-zero polynomial P) the finite sequence (a_0,....,a_n) where a_n is the "last" (largest n) non zero coefficient, or even only the finite subsequence of non-zero coefficients. (One usually says the zero polynomial has "all its coefficients equal to zero".) In particular, a polynomial does not have a unique representation. For clarity, if you are referring only to non-zero coefficients, and/or non-zero terms, you should say so.

You may appreciate Mike Artin's discussion, p.350-351 in his book Algebra, of this point, which he says "creates a nuisance".

 

[FactChecker]

IMO, it is fairly standard to call the constant term, $a_0$, of a polynomial the "zero'th coefficient" and the coefficient of $x^i$ is the "i-th coefficient, $a_i$".

 

[littlemathquark]

I can't answer these questions:
Let $P(x)=3x^5+x^2-4$
a) What's the product of coefficients of polynomial $P(x)$? $-12$ or $0$?
b)What's the first- degree term of the polynomial $P(x)$? $0$ or not exist?

 

[martinbn]

I can't answer these questions:
Let $P(x)=3x^5+x^2-4$
a) What's the product of coefficients of polynomial $P(x)$? $-12$ or $0$?
b)What's the first- degree term of the polynomial $P(x)$? $0$ or not exist?

Where are these questions from? A book? Lectures? Exam? You will need to follow the definitions from your source.

 

[littlemathquark]

İt is from my mind and not sure which definition must follow.

 

[martinbn]

İt is from my mind and not sure which definition must follow.

What do you mean by "must follow"? If it is a problem you made up, follow which ever one you want, or both.

 

[jackjack2025]

Why does it matter littlemathquark? What difficulty are you having?

The 'standard' or conventional answers would be
a) -12
b) degree is 5

As martinbn and others have said, if you want to define in a non-conventional way, so be it. But why? What are you trying to achieve?

 

[littlemathquark]

I try to achive true or logical answers. For b) I'm looking for #x^1# term if it's exist.

 

[jackjack2025]

I try to achive true or logical answers. For b) I'm looking for #x^1# term if it's exist.

Why are you looking for that? You can look for what you want, but are you trying to solve an actual problem?

If not, what is the point? What does it matter?

There are standard conventions used in mathematics, but mathematics works in this logical way
Definitions => Assumptions => Logical Consequences of those assumptions.

What is your definition of a polynomial, then what is your definition of the 'first degree' of that polynomial? Then you have what you want.

Or use the standard math conventions.

 

[martinbn]

I try to achive true or logical answers. For b) I'm looking for #x^1# term if it's exist.

I still have the feeling that you think that definitions represent absolute truth. But they can be, and are, different depending on the source or the person. For example some people exclude zero from the natural numbers some include it. You cannot ask a question that relies on whether zero is or is not a natural number and expect a definite answer without specifying which convention you use.

 

[FactChecker]

I can't answer these questions:
Let $P(x)=3x^5+x^2-4$
a) What's the product of coefficients of polynomial $P(x)$? $-12$ or $0$?
b)What's the first- degree term of the polynomial $P(x)$? $0$ or not exist?

a) IMO, if you want the product of the non-zero coefficients, say that.
Otherwise, I would assume that you are talking about the space of 5-th degree polynomials, $a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$ and asking what $a_0a_1a_2a_3a_4a_5$ is.

b) The coefficient of $x_1$ in that polynomial is defined and is 0.

 

[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.