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Math Help - Definition of a polynomial

  1. #1
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    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?
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  2. #2
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    Re: Definition of a polynomial

    Quote Originally Posted by AaPa View Post
    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 x^n,~x^{n-1},~x^{n-2} are,

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

    x^{n-1}\rightarrow a_1

    x^{n-2}\rightarrow a_2

    Notice that the subtrahend of the power of x becomes the subscript of a. For the last term the subtrahend is n (Since the variable in the last term is, x^{0}=x^{n-n}). Therefore the corresponding coefficient should be, a_n
    Thanks from AaPa
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  3. #3
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    Re: Definition of a polynomial

    thanks.
    please also tell me
    how do we express in this form

    x2 + 22
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  4. #4
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    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 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:

    p(x) = x^2+2^2 or
    p(x) = x^2+4
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  5. #5
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    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
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    Re: Definition of a polynomial

    Quote Originally Posted by AaPa View Post
    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, p(x) = a_{0}x^n+a_{1}x^{n-1} + a_{2}x^{n-2} +....+a_n with x^2-4. What are the values for the coefficients a_0,~a_1,~a_2,~\cdots~,a_n ?
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  7. #7
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    Re: Definition of a polynomial

    That is what I am not able to tell.
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  8. #8
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    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 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:

    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"):

    a_0 = 1, a_1 = 0, a_2 = -4.

    that is: x^2 - 4 = (1)x^2 + (0)x + (-4)
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  9. #9
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    Re: Definition of a polynomial

    Thank you.
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  10. #10
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    Re: Definition of a polynomial

    1 more thing.
    What do we call a polynomial that has more than 4 terms?
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  11. #11
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    Re: Definition of a polynomial

    Quote Originally Posted by AaPa View Post
    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
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