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Polynomial Equations : Solutions, Conjugates and Shift Operator

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Prove that if p is a polynomial with real coefficients, and if is a (complex) solution of P(E)z = 0, then the conjugate of z, the real part of z, and the imaginary part of z are also solutions.

Note: This is from a numerical analysis course, and here P(E) refers to a polynomial in E, the "shift operator" for a sequence.

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Solution Summary

Polynomials, complex solutions, conjugates and shift operators are investigated.

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Hi,
_
If z is a complex number, and z its conjugate, first you need to prove that conjugation distributes over multiplication and addition:
______ _ _
(z1 * z2) = z1 * z2
_______ _ _
(z1 + z2) = (z1) + (z2)

If z= x+iy, then the complex number x-iy is called the conjugate of the complext number z and is
_ __ __
written as z . It is easily seen that numbers conjugate to z1+z2 and z1z2 are (z1) + (z2) and
_ _
z1 z2 respectively.

Let P(x) = a x^n + a x^n-1 + a x^n-2 +...+ a x + a
...

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