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Thread: trouble with graphs of polynomials.

  1. #1
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    trouble with graphs of polynomials.

    so im having trouble understanding the concept of assuming the graphs of polynomials. i wish i had exmaple questions but i dont, so i thought someone here could help.

    things i dont understand:
    - how to tell the least degree of a polynomial function.( i originally thought that it was by how many roots there were but doesnt seem to work for me.)
    - how to tell whether the leading coefficient is positive or negative.
    any help is appreciated. thanks
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  2. #2
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    Re: trouble with graphs of polynomials.

    Quote Originally Posted by Recyclop View Post
    so im having trouble understanding the concept of assuming the graphs of polynomials. i wish i had exmaple questions but i dont, so i thought someone here could help.

    things i dont understand:
    - how to tell the least degree of a polynomial function.( i originally thought that it was by how many roots there were but doesnt seem to work for me.)
    - how to tell whether the leading coefficient is positive or negative.
    any help is appreciated. thanks
    Consider the general polynomial $\displaystyle a_nx^n + a_{n - 1}x^{n - 1} + ~...~ + a_1x + a_0$

    The least degree of the polynomial is the lowest power of x appearing in the polynomial. In this case 0. (Technically the last term of the polynomial can be written as $\displaystyle a_0 x^0$. )

    The leading coefficient is the coefficient of the highest power in the polynomial. In this case $\displaystyle a_n$.

    So if we have the polynomial $\displaystyle 3x + 5x^3 - 8x^4$ what is the least degree of this function? What is the leading coefficient?

    -Dan
    Last edited by topsquark; Dec 5th 2012 at 08:21 AM.
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  3. #3
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    Re: trouble with graphs of polynomials.

    I am not sure there is a universally accepted concept of "the least degree" of a polynomial. The degree of a polynomial is the highest degree of its terms, where the degree of a term $\displaystyle x^n$ is $\displaystyle n$. Thus, the degree of $\displaystyle a_nx^n + a_{n - 1}x^{n - 1} + ~...~ + a_1x + a_0$ is $\displaystyle n$.

    It's difficult to determine the degree from a graph when the axes are not graduated. For example, the graphs of $\displaystyle x^2$ and $\displaystyle x^4$ look similar. The number of real roots of a polynomial does not exceed its degree, but it can be strictly less than the degree.

    For any polynomial $\displaystyle f(x)$, $\displaystyle |f(x)|\to+\infty$ as $\displaystyle x\to\pm\infty$. You can tell whether the degree of $\displaystyle f(x)$ is even or odd by observing the behavior at ∞: if $\displaystyle f(x)$ tends to infinities of the same sign when x tends to +∞ and -∞, then the degree is even; if, for example, $\displaystyle f(x)\to+\infty$ as $\displaystyle x\to+\infty$ and $\displaystyle f(x)\to-\infty$ as $\displaystyle x\to-\infty$, then the degree is odd.

    The leading coefficient of $\displaystyle f(x)$ is positive iff $\displaystyle f(x)\to+\infty$ as $\displaystyle x\to+\infty$, and it is negative iff $\displaystyle f(x)\to-\infty$ as $\displaystyle x\to+\infty$.
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    Re: trouble with graphs of polynomials.

    @topsquark,
    i understand how to find those things while seeing the polynomial infront of me the problem is interpreting that infromation from a graph.
    for the polynomial you gave me the least degree would be 1, and the leading coefficient would be -8.

    @emakarov i think im starting to understand now, thanks i will continue to use your post as a reference as i study for my finals
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