Having problems with polynomial

[kingerd]

hi everyone new to the forum here.

here's my problem. I have to tell whether a given expression is a polynomial in x or not, and if so give its degree.

I've figured out 4 of the 5 but I'm stuck on the third one

2^x + 3x

since 2 is being raised to x and 3 is being multiplied by x I'm completely confused on how to determine if it is or not and what the degree would be if it has one

please help

 

[TD]

Well, what have you seen as definition of a polynomial?

 

[kingerd]

5x^2 - 3x + (1/2)

with the degree being 2 since the highest power of x is 2

 

[TD]

Yes, but that's an example.
What is the definition of a polynomial, have you seen that?
When do you call a mathematical expression a 'polynomial' (in x)?

 

[mladen]

no, it's not a polynomial if you ask me

 

[TD]

no, it's not a polynomial if you ask me

That's correct, but it would be good if kingerd is able to find out why that is.

 

[mladen]

bish bash bosh :)
because all parts must have an x to the power of a real number.

http://en.wikipedia.org/wiki/Polynomial

 

[kingerd]

oh sorry, you mean..

a polynomial in one variable is any expression of the type

anX^n + an-1X^n-1 + ... + a2X^2 + a1X + a0

where n is a nonnegative integer, an,...,a0 are real numbers called coefficients, and an not equal 0

 

[TD]

If that's the definition which you are using, then clearly your example can't be a polynomial since there's an x as power (exponent), you see?

 

[kingerd]

I'm sorry, but is that because since we don't really know the value of x it could be a non-negative number? I just want to make sure i understand it

 

[TD]

Oh no, even if we were sure that x was a positive integer, it still wouldn't be allowed to have x as an exponent. At least: it wouldn't be a polynomial in x anymore.

 

[sporkstorms]

I'm sorry, but is that because since we don't really know the value of x it could be a non-negative number? I just want to make sure i understand it

Let's start with this, a general form for polynomials:
$$a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0$$

What I mean by form, is that for an expression to be a polynomial, you need to be able to fit it into the above expression with the constraints you gave: n is any nonzero integer, a_n are real, etc.

Here's an example:
$$x^3 + 5x^2 + 3$$
We can plug it into the above form for a polynomial by saying: n = 3, a_3 = 1, a_2=5, a_1 = 0, a_0 = 3.

Take the expression 2^x and try to plug it into the polynomial form above. There's no way you can do it.. there are no (something)^x in there.

One additional condition that may have been taken for granted by your teacher, that i think might be causing a problem for you:
x (or whatever the variables may be) remains a variable. You won't typically plug in a numeric value for x and still call it a polynomial. If you took that 2^x and plugged in x=3, and then said 2^3 - you have just a number, 8.

I point this out because it sounds like you're trying to do that: plug in a value for 2^x, then try to fit the number into the polynomial (by saying, for example, x=3: 2^x = 2^3 which can be written as y^3 with y=2, so therefore it's a polynomial). That's going too far, and is wrong.

Just try to stick to that original polynomial form. x's stay x's (or another letter), a_n's are numbers, n's are non-negative integers, etc.