Does anyone know easy and common method of finding the roots of polynomial equation of any degree ?
My Math teacher said there was no general method but that : you will find that if you are given a polynomial function of the 3rd degree, usually there is one evident root such as 0, 1 or -1 that doesn’t require fancy math to prove. After that you know you can factor your polynomial function by (x-x1) where x1 is your solution. You get something like this : (x-x1)(ax^2+bx+c) You must then determine a, b, and c so that your function is equal to the first one given. Solving ax^2+bx+c is then a peace of cake, you combine those solutions with the first one and voilà. You can apply this by factoring several times to functions of a degree higher than the 3rd. Of course this is only true for a high school student like me…
Hey mate, there certainly exists a formula for the 3rd and 4th power (wikipedia search ‘cubic formula’ to get you started), however I don’t believe closed form solutions exist for orders greater than four. Generally the tactic employed is to either attempt to factor the nth order polynomial into 2,3th order polynomials and solve each component individually, i.e. if you had sum( a(i)x^(i) ; i = 0 to n) = 0 attempt to factor into form (a1x^2 + b1x + c)(a2x^2 + b2x + c2) …..( …) = 0 and using the property that if f(x) = a(x)b(x)c(x)……n(x) = 0 –> a(x) = 0 and/or b(x) = 0 and/or c(x) = 0 etc etc and solve each ‘easier component’ individually. This however can be a tiresome task, and generally it is common practise to employ a numerical approach (i.e. Newton-Rhapson Method) to computationally yield solutions to the expressions.
to do roots for powers higher you use synthetic division for example 2X^5 +X^4+X^3+X^2+X+1 you get all the factors for the constant +-1 then all the factors for teh first coefficent +-1+-2 the you put all the facorts for the constant over the first coefficent +-1 +-1/2 the you do synthetic division using ones that you think will work untill you get a quadratic then you either factor or use quadratic formula hope this helped
