Monday, January 28, 2008

monic polynomials

In today's blog, I will present a property of monic polynomials. A monic polynomial is a polynomial of degree n where the coefficient of xn is 1.

Lemma 1: Division by a monic polynomial

If f,g,h are polynomials such that f = g/h and g,h are monic. Then f is also monic.

Proof:

(1) Let g(x) = a0xr + a1xr-1 + ... + ar-1x + ar

(2) Let h(x) = b0xs + b1xs-1 + ... + bs-1x + bs

(3) Let f(x) = c0xt + c1xt-1 + ... + ct-1x + ct

(4) Since g(x) = h(x)*f(x), it follows that:

r = s + t

and

a0 = b0*c0

(5) Since a0 = b0*c0, it follows that:

c0 = a0/b0

(5) Since g(x) is monic and h(x) is monic, it follows that a0 = 1 and b0= 1.

(6) It therefore follows that f(x) is monic since:

c0 = a0/b0 = 1/1 = 1

QED

1 comment :

Moyayo said...

Well Done, and easy to understand!!