proof of factor theorem
Suppose that is a polynomial with real or complex coefficients of degree . Since is a polynomial, it is infinitely differentiable. Therefore, has a Taylor expansion about . Since , the terminates after the term. Also, the remainder of the Taylor series vanishes; i.e. (http://planetmath.org/Ie), . Thus, the function is equal to its Taylor series. Hence,
If , then . Thus, , where is the polynomial . Hence, is a factor of .
Conversely, if is a factor of , then for some polynomial . Hence, .
It follows that is a factor of if and only if .
|Title||proof of factor theorem|
|Date of creation||2013-03-22 12:39:54|
|Last modified on||2013-03-22 12:39:54|
|Last modified by||Wkbj79 (1863)|