Finding Real and Complex Zeroes of Polynomials

[m_s_a]

find all x

 

[rock.freak667]

Look up remainder and factor theorem

 

[HallsofIvy]

Also, look up the "rational root theorem": Any rational root of $a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot + a_1x+ a_0= 0$ must have denominator the divides $a_n$ and denominator that divides $a_0$. Here, $a_3= 1$ and $a_0= 4$ so there aren't too many possibilities.

 

[dynamicsolo]

No dice: except for the second equation, where you can pick out one "zero" right away, the candidates from the Rational Zeroes Theorem don't work here... (A graph of each suggests that those real zeroes that do exist appear to be resolutely irrational.)

What course is this for? I ask because these don't seem to be the sort of polynomials that factor nicely for an elementary course in algebra and functions. Are you allowed to use numerical methods? You might want to use the Intermediate Value Theorem to search for regions where the real zeroes exist and then use something like the Newton-Raphson method to home in on those real zeroes.

Are you also to find complex zeroes? It looks like you can extract the second real zero to solve the remaining quadratic equation. The situation with the first one looks like you may have a remaining quartic with four complex zeroes (the result from Descartes' Rule of Signs hints at this). Is there something we're supposed to notice about the coefficients that will help find those?