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Thread: Polynomial Degrees and Zeros

  1. #1
    Nov 2008

    Polynomial Degrees and Zeros

    Find a polynomial of degree 3 whose zeros are -3, 3/2 and 2


    I know C and B are not options because the degree is not three, but how do I figure out the rest?
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  2. #2
    Junior Member
    Nov 2008
    Do you know about the factorial decomposition of polynomials?
    If a polynomial $\displaystyle f(x)$ of degree 3 has 3 real roots, letīs say $\displaystyle \alpha, \beta,\gamma$ then you can write it down as $\displaystyle a(x-\alpha)(x-\beta)(x-\gamma)$ where $\displaystyle a$ is the leading coefficient of your polynomial.
    Obviously, this fact gets generalised for any polynomial of degree n with m roots, $\displaystyle n<m$.

    That means that you can write your polynomial as $\displaystyle a(x+3)(x-3/2)(x-2)$ since they didnīt tell you explicitly nothing about the leading coefficient of the polynomial, $\displaystyle a$ might be any real number different of zero. So set $\displaystyle a=2$ since all your options have L.C.=2.
    Apply distributive law (if you have trouble with that tell me and Iīll write it down) and you get option d.
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