Seems easy, but how would you show that every polynomial of fifth degree has at least one real root?
Thanks for any help.
Hello, OntarioStud!
How would you show that every polynomial of fifth degree has at least one real root?
Graphically, maybe?
Given: .$\displaystyle f(x) \:=\:ax^5 + bx^4 + cx^3 + dx^2 + ex + f$
For positive $\displaystyle a: \;\lim_{x\to\infty}f(x) \,=\,\infty$ and $\displaystyle \lim_{x\to-\infty}f(x) \,=\,-\infty$ *
Since the polynomial function is continuous, it must cross the x-axis somewhere.
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* .Of course, the argument hold for negative $\displaystyle a$ as well.
In complex analysis it is shown that any non-constant polynomial is reducible to linear ploynomials over the complex field. Furthermore, the real polynomials are reducible to linear factors such that they come in conjugate pairs of complex numbers. Since there is an odd number of factors it is not possible to a polynomial into non real complex numbers. Thus the factor $\displaystyle (x-r)$ appears where $\displaystyle r$ is real. This assures us that there is at least one real solution.