How can I factor higher order equations to find the poles and zeros?

[karaandnick]

For the life of me I cannot figure out how to factor higher order equations so that I can find the poles and zeros, my professor will not show how to do it and expects everyone to already know how, but i have forgotten and cannot find anywhere on the web to show a one and done method, please help!

here is one of the equations
x^6+24x^5+247x^4+1518x^3+5487x^2+10944x+8840

and another
x^3+3.46x^2+7.392x+1.7056

and 5th order and so on. i don't need these solved i just need someone to tell me how to go about solving them. is there any easy way that will work on any degree or will i have to apply multiple methods to each of them depending on the order.

 

[eumyang]

Look up Rational Zeros Theorem. Note that the first polynomial will not factor into irreducible polynomials with integer coefficients.

 

[HallsofIvy]

Your professor won't "show you how to do it" because there is NO general method for factoring higher degree polynomials. Factoring is pretty much "trial and error". The "rational zeros theorem" that eumyang mentioned is a good start: If x= m/n is a zero of the polynomial a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0, with all coefficients integers, then n must evenly divide the leading coefficient, a_n and m must divide the constant term, a_0. But even that just gives you some possible numbers to check- and there is no guarantee that a polynomial has rational zeros. If fact, there are polynomials, of degree 5 and higher, such that their zeros cannot be written in terms of radicals.