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) = a₀x^r + a₁x^r-1 + ... + a_r-1x + a_r

(2) Let h(x) = b₀x^s + b₁x^s-1 + ... + b_s-1x + b_s

(3) Let f(x) = c₀x^t + c₁x^t-1 + ... + c_t-1x + c_t

(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