Question 324129
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*[tex \Large \alpha] is a zero of a polynomial if and only if *[tex \Large x\ -\ \alpha] is a factor of of the polynomial.


Every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity.


Therefore the complete set of factors for the desired polynomial is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(x\  -\ (-5)\right)\left(x\  -\ (-5)\right)\left(x\  -\ (0)\right)]


All you need to do to specify one of the infinite set of polynomials that fits the requirement is to multiply:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(x\  +\ 5\right)\left(x\  +\ 5\right)\left(x\right)]


Just multiply it out.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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