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Question 650721: If f(x) is a polynomial function of degree greater than 2, and f(x) has only one zero, then can you say whether the zero is real or complex?
Please explain, I'm confused. Thanks.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the solution is to use descartes rule of signs.
the rational roots test helps to find the rational roots if they're real.
this reference talks about his rule.
it tells the maximum number of positive real roots of an equation and the maximum possible number of negative roots of an equation.
http://people.richland.edu/james/lecture/m116/polynomials/zeros.html
a very simple way to determine if the eqution has real roots and around where the real roots lie is to graph the equation.
where the equation crosses the x-axis is where the real roots are.
there's a whole list of operations to be performed to find the roots.
the following references address the procedures for doing that.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut38_zero1.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut39_zero2.htm
to solve your immediate problem, if the polynomial is of degree greater than 2 and it has only one zero, that must mean the zero is real because any polynomial has the number of roots that are equal to the degree of the polynomial.
not all of them are real.
at least some of them can be complex.
it can't have only 1 zero.
it can have only 1 real zero.
it can't have only 1 complex zero because complex zeroes, if they exist, come in conjugate pairs.
that last statement may be the answer you are looking for.
complex roots come in conjugate pairs therefore you can't have only 1 complex root to a polynomial.
long way to get there, but your answer lies in here somewhere.
here's a reference on complex roots.
http://www.themathpage.com/aprecalc/factor-theorem.htm
you have to get through the rest of it, but there is a section in there that explains that complex roots always come in conjugate pairs.
Answer by ikleyn(53153)  (Show Source):
You can put this solution on YOUR website! .
If f(x) is a polynomial function of degree greater than 2, and f(x) has only one zero,
then can you say whether the zero is real or complex?
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As the problem is worded, it only makes sense, when the coefficients of the polynomial
are real numbers (the condition which is omitted in the problem).
It is because if the polynomial has complex number coefficients, then
the set of the roots must be considered over complex numbers, and in
this case the number of roots is equal to the degree of the polynomial,
which in this problem is greater than 2.
OK. But then, if the existing root is a complex number, then there is another complex root number
different from the first one and conjugated with the first complex number root.
But because we are given that f(x) has only one root, it means that the existing single
root is a real number.
Solved and explained.
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