# Thread: Definition of a polynomial

1. ## Definition of a polynomial

A polynomial is an expression p(x) of the form : p(x) = a0xn+ a1xn-1 + a2xn-2 +....+an

where

a1,a2......are real numbers

andn is whole no.

Q1. Why did we use ....n... as a subscript for .....a.... at the last?

2. ## Re: Definition of a polynomial

Originally Posted by AaPa
A polynomial is an expression p(x) of the form : p(x) = a0xn+ a1xn-1 + a2xn-2 +....+an

where

a1,a2......are real numbers

andn is whole no.

Q1. Why did we use ....n... as a subscript for .....a.... at the last?
Hi AaPa,

Notice that this is a nth degree polynomial. Hence there are n number of terms. The coefficients corresponding to $\displaystyle x^n,~x^{n-1},~x^{n-2}$ are,

$\displaystyle x^n=x^{n-0}\rightarrow a_0$

$\displaystyle x^{n-1}\rightarrow a_1$

$\displaystyle x^{n-2}\rightarrow a_2$

Notice that the subtrahend of the power of $\displaystyle x$ becomes the subscript of $\displaystyle a$. For the last term the subtrahend is $\displaystyle n$ (Since the variable in the last term is, $\displaystyle x^{0}=x^{n-n}$). Therefore the corresponding coefficient should be, $\displaystyle a_n$

3. ## Re: Definition of a polynomial

thanks.
how do we express in this form

x2 + 22

4. ## Re: Definition of a polynomial

I hope that I am understanding your question correctly.

p(x) = a0xn+ a1xn-1 + a2xn-2 +....+an

This is a polynomial function.

Notice that the expression $\displaystyle x^2+2^2$ looks similar to the right side of the equation you posted. It is a polynomial already, but it is not a polynomial function. If you wanted to express it as a function, you could write something like:

$\displaystyle p(x) = x^2+2^2$ or
$\displaystyle p(x) = x^2+4$

5. ## Re: Definition of a polynomial

I want to know how to substitute the values in p(x) = a0xn+ a1xn-1 + a2xn-2 +....+an

to get

x2 - 4

6. ## Re: Definition of a polynomial

Originally Posted by AaPa
I want to know how to substitute the values in p(x) = a0xn+ a1xn-1 + a2xn-2 +....+an

to get

x2 - 4
Compare the standard form of the nth degree polynomial, $\displaystyle p(x) = a_{0}x^n+a_{1}x^{n-1} + a_{2}x^{n-2} +....+a_n$ with $\displaystyle x^2-4$. What are the values for the coefficients $\displaystyle a_0,~a_1,~a_2,~\cdots~,a_n$ ?

7. ## Re: Definition of a polynomial

That is what I am not able to tell.

8. ## Re: Definition of a polynomial

one small correction: a n-th degree polynomial contains n+1 terms, n terms for each power of x, and the constant term.

the degree of a polynomial p(x) is the "exponent number" of the highest power of x that occurs in it. for example, in $\displaystyle x^2 - 4$, the highest power of x that occurs is 2 (in x squared), so this polynomial is of degree 2. so n = 2.

if we write this in the form:

$\displaystyle x^2 - 4 = p(x) = a_0x^2 + a_1x + a_2$

we get the following values for the coefficients (this is the proper term for the "a's"):

$\displaystyle a_0 = 1, a_1 = 0, a_2 = -4$.

that is: $\displaystyle x^2 - 4 = (1)x^2 + (0)x + (-4)$

Thank you.

10. ## Re: Definition of a polynomial

1 more thing.
What do we call a polynomial that has more than 4 terms?

11. ## Re: Definition of a polynomial

Originally Posted by AaPa
1 more thing.
What do we call a polynomial that has more than 4 terms?
Read this and you will find the answer: Polynomial - Wikipedia, the free encyclopedia