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.
(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
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