Is there another way to find the roots of 3rd degree and higher polynomials instead of using trial and error (using synthetic division)?
One way it helps to look at roots of 3rd degree is that it's the same as the what is under the root to the power of (1/3).
Ex:
(3√x)= x^(1/3)
I would then do this with a higher polynomial like
(3√x^5)= (x^5)^(1/3)
= x^(5/3)
I use Newton's method, which involves repeated iterations, where you start with some reasonable guess and within a few steps it will "Zoom in" on the answer
in general it says
if for some function f(x) , starting with a guess of x
new x = x - f(x)/derivative of f(x)
so for a cubic
f(x) = ax^3 + bx^2 + cx + d
then
newx = (2ax^3 + bx^2 - d)/(3ax^2 + 2bx + c)
the result of newx then becomes the input for x in the second round.
Your newx will get closer and closer to your input value of x
You will be amazed how fast it is.
BTW, no matter how 'weird' your first guess is, eventually it will get you there, it just takes longer.
BTW#2
for square roots, eg. √5 it would work this way
newx = (x^2+5)/(2x)
guess at x = 2
newx - x
---- 2.25
2.25 ---2.236111
2.236111 -- 2.2360679
2.2360679 -- 2.2360679 same as input
4 steps to get 7 decimal accuracy !!!
try √5 on your calculator
For polynomials of third and higher degree, it matters whether you are looking for an exact solution (factorable) or contented with a numerical answer.
For factorable polynomials, apart from testing by trial and error for integer roots, you can read up on Descartes rule of signs to help you limit the number of trials.
See:
http://en.wikipedia.org/wiki/Descartes%27_rule_of_signs
There are many tools for numerical solutions. One of them is graphing, another is by iteration. One of the well-known iteration mthods is Newton's method. See for example:
http://en.wikipedia.org/wiki/Newton%27s_method
There is an analytical way of getting a exact closed-form solutions to cubic equations, but it fills two pages. For higher order polynomials, I am not aware of any exact methods.
Yes, there are other methods to find the roots of 3rd-degree and higher polynomials besides trial and error or synthetic division. One of the most commonly used methods is the Rational Root Theorem combined with synthetic division.
The Rational Root Theorem states that if a polynomial has a rational root (a root that can be expressed as a fraction), it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
To apply this theorem and find potential rational roots, you can start by listing all the possible factors of the constant term and the leading coefficient. Then, you use synthetic division to test each potential root until you find one that results in a remainder of zero. Once you find a root using synthetic division, you can reduce the polynomial by dividing it by (x - root). This process can be repeated on the reduced polynomial until you find all the roots.
Alternatively, you can also use graphing calculators or computer software to graph the polynomial and visually determine any real roots. This method can be helpful in identifying approximate roots, although it may not be as precise as algebraic methods.
It is worth mentioning that finding roots of polynomials with degrees higher than three can become increasingly difficult and may require more advanced methods or numerical approximation techniques such as Newton's method or the use of software specifically designed for solving higher-degree polynomials.