Roots of a polynomial equation
Posted by Jonathan Celis
Reading i have found an interesting subject for numerical methods, and its called Roots of equation, and i am going to talk about polynomial equations. Here are three important theorems relating to the roots of a polynomial:
1.)A polynomial of n-th degree can be factored into n linear factors.
2.)A polynomial equation of degree n has exactly n roots.
3.)If (x − r) is a factor of a polynomial, then x = r is a root of the associated polynomial equation.
The cubic polynomial f(x) = 4x3 − 3x2 − 25x − 6 has degree 3.
This polynomial can be factored and written as
4x3 − 3x2 − 25x − 6 = (x − 3)(4x + 1)(x + 2)
So we see that a 3rd degree polynomial has 3 roots.
The associated polynomial equation is formed by setting the polynomial equal to zero:
f(x) = 4x3 − 3x2 − 25x − 6 = 0
In factored form, this is:
(x − 3)(4x + 1)(x + 2) = 0
We see from the expressions in brackets and using the 3rd theorem from above, that here are 3 roots, x = 3,-1/4,−2.
In this example, all 3 roots of our polynomial equation of degree 3 are real.
Since (x − 3) is a factor, then x = 3 is a root.
Since (4x + 1) is a factor, then x =-1/4 is a root.
Since (x + 2) is a factor, then x = −2 is a root.
The equation x5 − 4x4 − 7x3 + 14x2 − 44x + 120 = 0 can be factored and written as:
(x − 2)(x − 5)(x + 3)(x2 + 4) = 0
We see there are 3 real roots x = 2, 5, -3, and 2 complex roots x = ±2j, (where j = √-1).
So our 5th degree equation has 5 roots altogether.
About complex roots, the following theorem applies :
If the coefficients of the equation f(x) = 0 are real and a + bj is a complex root, then its conjugate a − bj is also a root.
So here its an example:
The factors of the polynomial x3+ 7x2 + 17x + 15 are found using LiveMath:
x3 + 7x2 + 17x + 15 = (x + 3)(x + 2 − j)(x + 2 + j)
So the roots are
x = −3
x = −2 + j and
x = −2 − j
There is one real root and the remaining 2 roots form a complex conjugate pair.
I hope you enjoyed this, and here and interesting link for you to see about newton-raphson method.
http://www.youtube.com/watch?v=lFYzdOemDj8
