Given:
#f(x) = x^8+3x^2-4#
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Fundamental Theorem of Algebra
Since this polynomial is of degree #8# ("octic"), the Fundamental Theorem of Algebra allows us to deduce that it has exactly #8# complex (possibly real) zeros counting multiplicity.
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Descartes' Rule of Signs
Note that the pattern of signs of the coefficients of #f(x)# is #+ + -#. With one change of sign, Descartes' Rule of Signs allows us to deduce that #f(x)# has exactly one positive real zero.
The pattern of signs of #f(-x)# is also #+ + -#. Hence we can deduce that #f(x)# has exactly one negative real zero.
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Conclusion
So two of the #8# zeros are real and #6# non-real.
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Bonus
Note that the sum of the coefficients of #f(x)# is zero. That is:
#1+3-4 = 0#
Hence we can deduce that #x=1# is a zero and #(x-1)# a factor. Since all of the terms are of even degree, #x=-1# is also a zero and #(x+1)# a factor.
