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

[littlemathquark]

TL;DR

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.