Question 1027752
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The Fundamental Theorem of Algebra:  every non-zero, single-variable, degree *[tex \Large n] polynomial with complex coefficients has, counted with multiplicity, exactly *[tex \Large n] roots.


The Complex Conjugate Root Theorem:  if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a &#8722; bi is also a root of P.


A corollary to the FTA is that if *[tex \Large \alpha] is a zero of a single-variable polynomial with real coefficients, then *[tex \Large (x\ -\ \alpha)] is a factor of the polynomial.


Assuming real coefficients, then a representation of a third-degree polynomial with zeros -2, 2, and 4 would be:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ a(x\ +\ 2)(x\ -\ 2)(x\ -\ 4)]


Which has *[tex \Large x]-intercepts of *[tex \Large (-2,0)], *[tex \Large (2,0)], and *[tex \Large (4,0)], a *[tex \Large y]-intercept that is the additive inverse of the product of the zeros and the common coefficient *[tex \Large a], in this case *[tex \Large 16a], and opposite behavior as *[tex \Large x] decreases without bound from the behavior as *[tex \Large x] increases without bound.


Example for *[tex \Large a\ =\ 1] (magenta curve) and *[tex \Large a\ =\ -\frac{1}{2}] (blue curve).



*[illustration 3rdDegreePolyFunctions.jpg]


b) is clearly impossible because a polynomial function with four zeros, must by the FTA, be a 4th degree polynomial.


c) is impossible IF you are restricted to real coefficients.  The Complex Conjugate Theorem demands that complex roots of polynomial functions with real coefficients come in pairs, and the FTA demands exactly 3 roots.  Therefore, any 3rd degree polynomial function with real coeficients must have either 3 real roots or 1 real root and a conjugate pair of complex roots.  On the other hand, in the highly unlikely event that your instructor is having you consider complex polynomials, then c) is possible but well beyond my capability to illustrate it.


d) Try *[tex \Large -2], *[tex \Large 1\ +\ 2i], and *[tex \Large 1\ -\ 2i]


I'll leave verifying the expansion as an exercise for you, but the polynomial comes out to be:



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ x^3\ +\ x\ +\ 10]



*[illustration 3rdDegreePolyFunctionW2ComplexRoots.jpg]



John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  

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