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Math Help - Polynomial definition (no equation)

  1. #1
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    Polynomial definition (no equation)

    Hi

    My class is getting introduce to polynomial, and to start off we have an assessment on describing what a polynomial is and how it work.

    I'm having a little trouble finding the explaination on google.

    Can you guys assist?

    Questions are

    What patterns arise for polynomials of the form y = axn (called monomials)?
    What happens when the leading coefficient of a polynomial is changed?
    What happens when the constant term is changed?
    Is there are connection between the number of roots and the order of the polynomial?
    In there a connection between the number of turning points and the order of the polynomial?
    What if the polynomial equations are written in factorised form? What information do the factors give?

    Thanks in advance.
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  2. #2
    MHF Contributor
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    Re: Polynomial definition (no equation)

    Hey vjnr.

    Basically a polynomial is a function that takes an input x and produces an output y where for each x we get:

    y(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_0 where a_i's are just constant real numbers and n is a positive whole number (greater than 0).

    We can plot the function by calculating y(x) for a lot of different values of x and then by plotting the points either on graph paper or on a computer screen.

    The constant term is the a_0 term, the degree is the value of n, and monomials have a_n = not zero and the rest equal to zero.
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  3. #3
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    Re: Polynomial definition (no equation)

    Your questions can most easily be answered by graphing the polynomials.

    1. Graph y = x, x^2, x^3, x^4, x^5, x^6. This should answer #1.
    2. Graph y = 2x^3. Then graph y = -x^3, 5x^3, 0.1x^3, -4x^3 and note differences.
    3. Graph y = x^2 + 1, x^2 + 5, x^2 - 4 and note differences.
    4. Remember y = x^3 has order 3 and see the relationship between this and the number of roots.
    5. Remember, turning points are shifts from where the polynomial was increasing (or decreasing) to decreasing (or increasing). See if there is a connection.
    6. Graph a couple. Note all the characteristics of them. Try y = (x - 1)(x - 3).

    Math is much more fun when you discover these concepts yourself!
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