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Homework Help: Having problems with polynomial

  1. Jan 28, 2006 #1
    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:cry:
     
  2. jcsd
  3. Jan 28, 2006 #2

    TD

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    Well, what have you seen as definition of a polynomial?
     
  4. Jan 28, 2006 #3
    5x^2 - 3x + (1/2)

    with the degree being 2 since the highest power of x is 2
     
  5. Jan 28, 2006 #4

    TD

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    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)?
     
  6. Jan 28, 2006 #5
    no, it's not a polynomial if you ask me
     
  7. Jan 28, 2006 #6

    TD

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    That's correct, but it would be good if kingerd is able to find out why that is.
     
  8. Jan 28, 2006 #7
  9. Jan 28, 2006 #8
    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
     
  10. Jan 28, 2006 #9

    TD

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    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?
     
  11. Jan 28, 2006 #10
    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
     
  12. Jan 28, 2006 #11

    TD

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    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.
     
  13. Jan 28, 2006 #12
    Let's start with this, a general form for polynomials:
    [tex]a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0[/tex]

    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:
    [tex]x^3 + 5x^2 + 3[/tex]
    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.
     
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