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Math Help - Find All Zeros (real and imaginary)

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    Post Find All Zeros (real and imaginary)

    I need to use the given zero to find all the zeros real or imaginary of the function and also write f as a product of linear factors.

    Zero: 5-i

    F(x)=(3x^6)-(43x^5)+(218x^4)-(428x^3)+(156x^2)+300x-104
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    Quote Originally Posted by Kairo View Post
    I need to use the given zero to find all the zeros real or imaginary of the function and also write f as a product of linear factors.

    Zero: 5-i

    F(x)={3x^6}-{43x^5}+{218x^4}-{428x^3}+{156x^2}+300x-104
    If 5-i is a root of f(x), then 5+i also a root. Thus \left[z-(5-i)\right] and \left[z+(5-i)\right] are factors of f(x). Divide f(x) by x^2-10 x+26, and then you will get the quadric 3x^4-13x^3+10 x^2+10x-4. By plugging in possible integer roots, find that x = 2 yields the quadric zero, so x-2 is a factor of it. Divide the quadric by x-2, and you will get 3x^3-7x^2-4x+2. Keep going like this and find the roots of the cubic. Writing the equation as the product of linear factors means writing [LaTeX ERROR: Convert failed] as (x-a)(x-b)(x-c)(x-d)(x-e)(x-f), where a, b, c, d, e, and f are the roots you were asked to find, of which we already have three: 5-i, 5+i, and 2, and the other three are the roots of the cubic 3x^3-7x^2-4x+2 = 0.
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