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A part of the definition of a polynomial is :

$f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x^1+a_0$

where $a_n ,a_{n-1}, \dots, a_2, a_1, a_0$ are constants.

$\textbf{I have been confused as to why we have n.So if n is 5 do we have the following:}$ $$a_5x^5+a_4x^4+a_1x^1+a_0x^0$$

Please can I have help in understanding this ?

What in the world is $a_nx^n$ ?

J. M. ain't a mathematician
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Bahar
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    When $n=5$ you have $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x^1+a_0x^0$ – kingW3 Sep 03 '17 at 15:44
  • But what exactly does a_5 mean ? – Bahar Sep 03 '17 at 15:45
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    It's a particular coefficient depending on what the polynomial is, if it's $x^5+x+1$ for example then $a_5=1$. – kingW3 Sep 03 '17 at 15:46
  • That's interesting. :) Thank you. Shall I delete the question now ? – Bahar Sep 03 '17 at 15:47
  • $n$ is just a general place holder, so that the polynomial can be of arbitrary degree. – Dave Sep 03 '17 at 15:49
  • Some definitions don't have any $n$ and say that polynomials are $a_0+a_1x+a_2x^2+...$ in which only finitely many $a_i$ are non-zero. When they want to get the definition of degree faster they might stop the summation at the largest $n$ with $a_n\neq0$. In those cases they define polynomials as $a_0+a_1x+...+a_nx^n$ with $a_n\neq0$. – Hellen Sep 03 '17 at 15:49
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    No need to delete. Other people might have this same question, and this will be searchable for them. :) – G Tony Jacobs Sep 03 '17 at 15:49
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    Related: https://math.stackexchange.com/questions/2185587/what-actually-is-a-polynomial/2185648#2185648 – Ethan Bolker Sep 03 '17 at 15:53

2 Answers2

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The subscript on the coefficients is just a way of labeling them. The meaning of "$a_5$" is, "the coefficient of the degree $5$ term".

For example, consider the third degree polynomial:

$$5x^3-11x^2 + 9$$

In this case, we have $n=3$, because the degree is $3$. The coefficients are: $a_3=5, a_2=-11, a_1=0, a_0=9$. Thus, the $a$ notation is just a clear way of referring to each of the coefficients.

Does that help?

G Tony Jacobs
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  • Yes that does thank you. – Bahar Sep 03 '17 at 15:49
  • Shall I delete the question? I don't want to get down-voted. – Bahar Sep 03 '17 at 15:49
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    @Bahar No, don't delete questions after asking them. Press the up arrow next to any answers that are helpful and then press the checkmark to "accept" what you think is the best answer. – Stella Biderman Sep 03 '17 at 15:51
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    I think it's good to keep questions around. Someone else might be confused about the same thing, and this will help them. The point of Stack Exchange is really to build up a searchable database of questions and answers. I've upvoted your question, because I know you're not the only one who wonders about this. – G Tony Jacobs Sep 03 '17 at 15:51
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In mathematics we often us $n$ to describe a template. The expression you've written is the form that all polynomials have (though you're missing the term $+a_0x^0$).

So $x^3-2x^2+0x+1$ is a polynomial with $n=3$ and $x+1$ is a polynomial with $n=1$. Often times we drop terms with a coefficient of $0$, but I've included it to make the template clearer.

The general form for $n=5$ is $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x^2+a_0x^0$.

Stella Biderman
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  • Okay so like a binomial and trinomial and stuff thanks you. – Bahar Sep 03 '17 at 15:51
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    @behar yes, binomials and trinomials are examples of polynomials. "bi" means "two" and "tri" means "three," while "poly" means "many." So a binomial has two numbers (coefficents) a trinomial has three, and a polynomial can have any number (2, 3, 5, 100001) – Stella Biderman Sep 03 '17 at 15:53