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I've read the definition of a polynomial on Wikipedia, and got quite a different understanding from what was explained to me in a Khan Academy video.

In the Khan Academy video it says that:

$6$

is a polynomial, specifically a monomial because it's the same as:

$6x^0$

I was wondering if this was true, and also whether:

$6 + 1$

is a binomial or

6 + 15 - 2

is a trinomial

The Wikipedia articles defines a polynomial as an:

...expression consisting of variables (also called indeterminates) and coefficients...

And gives examples of:

$x^2 − 4x + 7$ and $x3 + 2xyz2 − yz + 1$

Something like I've shown higher up in my question, such as adding or subtracting 3 integers doesn't look like your typical polynomial. If 6 is a monomial and therefore a polynomial, then also 6 + 15 - 2 is a trinomial and polynomial, even if there are no explicitly written variables or indeterminates, as Wikipedia refers to them?

Zebrafish
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    No, $6+1=7$ is $7x_0$ as before, and $6+15-2=19$ is $19x_0$. – Dietrich Burde Mar 13 '19 at 23:08
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    A polynomial can also be constant, so $6$ is a polynomial. – Peter Mar 13 '19 at 23:10
  • @Dietrich Oh I see, 6 is a polynomial and 1 is a polynomial, but 6 + 1 isn't because it can be simplified and combined? – Zebrafish Mar 13 '19 at 23:11
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    It is still a polynomial, but not a binomial – Peter Mar 13 '19 at 23:12
  • @Peter Oh, so 6 + 15 - 2 can be simplified or combined into one coefficient/constant, and is therefore a monomial and polynomial. Thank you. – Zebrafish Mar 13 '19 at 23:14
  • @Zebrafish Exactly – Peter Mar 13 '19 at 23:15
  • You should specific the variables, e.g. $2^{\large 2x}$ is not a polynomial in $x$ but it is a polynomial in $y =2^{\large x}$, viz. $y^{\large 2}$ – Bill Dubuque Mar 13 '19 at 23:18
  • Related, possibly helpful: https://math.stackexchange.com/questions/2185587/what-actually-is-a-polynomial/2185648#2185648 – Ethan Bolker Mar 13 '19 at 23:26
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    I would say $6+1$ is a monomial. The same as $7x^0$. A polynomial has one coefficient for a power. If you have two for the same power, say $5x + x^2 + 3x$ then they are added and combined to one coefficient. So $5x + x^2 + 3x$ is the same as $x^2 + (5+3)x$. I, personally would say "$5x + x^2 + 3x$ is a polynomial" but I'd also say "it is of degree $2$ and it has two non-zero coefficients: $1$ is the coefficient of $x^2$ and $5+3$ is the coefficient of $x$". That's waht i'd say. Your instructor may say otherwise. – fleablood Mar 13 '19 at 23:29

1 Answers1

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Regarding the one variable case, if you want to be extra rigorous, one can define polynomials as lists of numbers $(a_0,a_1, \dots)$ such that the $a_i$ eventually start to be all zero. This is just a fancy way of saying that $3$ can be defined as $(3,0,0,\dots)$ and $3X+X^5$ as $(0,3,0,0,0,1,0,0,\dots)$, i.e. grouping all the coefficients that belong to a same power together.

What this means is that even if we informally have different terms with the same variable, at the time of assessing certain properties of a polynomial, one should write it into this "canonical" form. For example, we could write $7$ as $6+1$ or $X^2 + 4X$ as $X^2 + X + 3X$ or even $X(X+4)$. But $6+1$ is a monomial, because after regrouping, it corresponds to $(7,0,0, \dots)$. In the same way, $X^2 + X + 3X$ consists of two monomials, because it corresponds to $(0,4,1,0,0,\dots)$.

qualcuno
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    We could also say that $6+1=6X^0 + 1Y^0$. – enedil Mar 14 '19 at 00:03
  • Yeah, I avoided the multivariate case explicitly but still, one could index $\mathbb{N}_0^k$ for $R[X_1,\dots,X_k]$ as a countable list and regroup coefficients that way, so that in your example these belong to the same group (namely the place of the zero $k$-uple). – qualcuno Mar 14 '19 at 02:07