ATAR Notes: Forum
VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Martoman on March 02, 2010, 07:47:49 pm
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Find a polynomial equation of the lowest possible degree having roots of 1+2i and -3 if the coefficients are real and complex.
I get the complex one easily, multiply the roots in factored form then you get a polynomial degree 2 with complex coefficients. The real part is annoying me, I've attempted to sub in to the general 
but that just becomes ewwwy as i need more information for the three unknowns.
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complex conjugates...
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If the coefficients are real, then the complex roots occur in conjugate pairs. ie if 1+2i is a root, then so is 1-2i.
Hence p(z) = (z-1-2i)(z-1+2i)(z+3)
= (z^2-2z+5)(z+3)
= z^3+z^2-z+15
Therefore the equation is z^3+z^2-z+15=0
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arg.. fuck... thanks guys. Here is a good example of NOT reading the question. Here I was confounded by the fact that we couldn't possibly have only 2 solutions if its real and the highest power was z^2.... maybe i should listen to my intuition more... or learn to read English.