# A question about the definition of polynomials.

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 $a_nx^n$ ?

• 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 3 '17 at 15:44
• But what exactly does a_5 mean ? – Bahar Sep 3 '17 at 15:45
• 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 3 '17 at 15:46
• No need to delete. Other people might have this same question, and this will be searchable for them. :) – G Tony Jacobs Sep 3 '17 at 15:49
• – Ethan Bolker Sep 3 '17 at 15:53

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?

• Yes that does thank you. – Bahar Sep 3 '17 at 15:49
• Shall I delete the question? I don't want to get down-voted. – Bahar Sep 3 '17 at 15:49
• @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 3 '17 at 15:51
• 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 3 '17 at 15:51

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 $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x^2+a_0x^0$.

• Okay so like a binomial and trinomial and stuff thanks you. – Bahar Sep 3 '17 at 15:51
• @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 3 '17 at 15:53