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Show that complex conjugate is also a root of polynomial with real coefficients

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Muffins
#1
Oct5-08, 12:51 PM
P: 6
1. The problem statement, all variables and given/known data
Suppose that f(x) is a polynomial of degree n with real coefficients; that is,

f(x)=a_n x^n+ a_(n-1) x^(n-1)+ …+a_1 x+ a_0, a_n,… ,a_0∈ R(real)

Suppose that c ∈ C(complex) is a root of f(x). Prove that c conjugate is also a root of f(x)

2. Relevant equations

(a+bi)*(a-bi) = a^2 + b^2 where a, and b are always reals?
Not really sure if this helps or not.

3. The attempt at a solution

I'm really clueless on how to start approaching this. I was thinking perhaps the fundamental theorem of algebra might be of some use, or perhaps the fact that a number of complex form multiplied by it's conjugate is a real number, but I'm really not sure.

Could anyone give me a nudge in the right direction?
Any help would be greatly appreciated!! Thanks!
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Dick
#2
Oct5-08, 02:53 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
The root is where f(c)=0. Take the complex conjugate of that equation.
Muffins
#3
Oct5-08, 02:54 PM
P: 6
Never mind! I found the actual theorem on the web, and I think this is pretty much what I was looking for anyway

Consider the polynomial
f(x)=a_n x^n+ a_(n-1) x^(n-1)+ …+a_1 x+ a_0, a_n,… ,a_0∈ R(real)
where all a*x are real. The equation f(x) = 0 is thus
a_n x^n+ a_(n-1) x^(n-1)+ …+a_1 x+ a_0, a_n = 0
Given that all of the coefficients are real, we have
a_n x^n(conjugate) = a_n x^n(x is conjugate)
Thus it follows that
a_n x^n(conj)+ a_(n-1) x^(n-1)(conj)+ …+a_1 x(conj)+ a_0, a_n = 0(conj) = 0
and thus that for any root ζ its complex conjugate is also a root.

Muffins
#4
Oct5-08, 02:55 PM
P: 6
Show that complex conjugate is also a root of polynomial with real coefficients

Hahah, thanks Dick! I caught on a little late, but thanks a bunch for your reply!


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