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I am studying Polynomial functions and their Graphs.

I am currently looking at the definition for a polynomial function and I am trying to arrive at a deeper understanding; thus, please excuse questions that seem obvious.

Nonetheless,

A polynomial function of degree $n$ (What is $n$, a arbitrary variable chosen?) is a function of the form:

$$ P(x) = a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0 $$

where $n$ is a nonnegative integer and $a_n$ does not equal 0.

The numbers $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are called the coefficients of the polynomial.

The number $a_0$ is the constant coefficient or constant term.

The number $a_n$, the coefficient of the highest power, is the leading coefficient, and the term $a_n x^n$ is the leading term.

Questions:

1) What do the subscripts indicate, such as $n$?

2) In algebra, I learned that constants are for example $1, 2, 3, 4$; however, they describe constants as constant coefficients in the book. Can someone explain the reason behind that?

As you can see, I am new to learning mathematics, so please be simple.

Prologue
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3 Answers3

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Here is a concrete example which you can compare to the definition to perhaps gain some understanding.

The following expression is a polynomial of degree 5,

$$ 3x^5-2x^4+x^2-x+2.$$

When I say it has a degree of 5 all I mean is that the highest power of $x$ which you will find in the expression is the 5th power.

The coefficients in this polynomial are 3,-2,0,1,-1, and 2. Since each power of $x$ has its own coefficient it is sometimes useful to give them the following labels.

$$ a_5 = 3$$ $$ a_4 = -2$$ $$ a_3 = 0$$ $$ a_2 = 1$$ $$ a_1 = -1$$ $$ a_0 = 2$$

Notice that each coefficient has as a subscript the power of $x$ that the coefficient is multiplied. This is just a notational convention for labeling or organizing coefficients and it isn't really mathematically important, but it is useful to understand.

It may seem odd to you that I have a coefficient $a_3=0$ listed. This is just there to indicate that our polynomial doesn't have an $x^3$ term.

Spencer
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  • Thank you. However, what does the x^n indicate if subscript "n" adjacent to the "a" represents the degree? – Prologue Apr 05 '14 at 19:45
  • If the degree is $n$ then the highest power of $x$ is $x^n$. The subscript on the $a$ just refers to which power of $x$ it belongs to. If I write $a_n$ then I am referring to the coefficient of the $n$'th power of $x$. Notice that in the definition the subscript always agrees with the power of $x$. – Spencer Apr 05 '14 at 19:52
  • In our example the highest power of $x$ was $x^5$. This indicates that the degree of the polynomial is $5$. The coefficient of the $x^5$ term is $3$. Because the $3$ is multiplying $x^5$ is can call it $a_5$. The reason the subscript is a $5$ has nothing in particular to do with the degree, but just the fact that the power of $x$ it was attached to was a $5$. – Spencer Apr 05 '14 at 19:55
  • When you have a particular polynomial it isn't. But if you are trying to say something about all possible polynomials you have to be more abstract and not commit yourself to a choice of coefficients. When doing this there are going to be $n$ different coefficients and its helpful to distinguish them by giving each one a different subscript. – Spencer Apr 05 '14 at 20:04
  • You may have seen that the generic equation for a line is $y=mx+b$. We could write this instead as $y=a_1 x + a_0$. The mathematical content obviously doesn't change. When you are working with a polynomial that has a large degree (or even an unspecified degree) the subscripts make it easier to keep track of things. – Spencer Apr 05 '14 at 20:07
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1) The subscripts are an easy and clever way to denote the coefficients of every term. For example, one could write a polynomial function as: $$f(x)=\alpha x^n+\beta x^{n-1}+\gamma x^{n-2}+\zeta x^{n-3}+\xi x^{n-4}+\cdots+\eta$$ But if I ask you for example: what is the coefficient of the term of degree $n-5$, you'll have to search and look for the symbol and recognize its name... It also happens that we don't have infinitely many symbols so you'll have a hard time writing the standard form of a polynomial with coefficient $200$ for instance. That's why the $a_{\displaystyle\color{grey}{\text{subscript}}}$ notation was adopted. So when you write the standard form and I ask you to find the coefficient of the term with degree $n-30$ you'll easily respond: $a_{n-30}$.

2) Why we say that they are constant coefficients is that you can view them as a constant times a variable: $$f(x)=a^nx^n+\cdots+a_1x+\color{blue}{a_0x^0}.$$

I hope this helps.
Best wishes, $\mathcal H$akim.

Hakim
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It is just to make the writing more general. As you wrote $$P_n(x)=a_n x^n+a_{n-1} x^ {n-1}+a_{n-2} x^{n-2}+...+a_{2} x^2+a_1 x+a_0$$ The subscript $k$ assigned to the coefficient is the one associated to power $k$ of the variable. You will see soon that, because of the generalization, we can shorten the writing to $$P_n(x)=\sum _{k=0}^n a_k x^k$$

When you will have a practical problem, the coefficients $a_k$ will be replaced by numbers.

Is this making things clearer to you ? If not, please post.

Claude Leibovici
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