Let fx and gx be polynomials in Fx with gx 0 Show that if fx

Let f(x) and g(x) be polynomials in F[x], with g(x) 0. Show that if f(x)| g(x), then deg f(x) g(x).

Solution

given f(x) and g(x) are polynomials in F[x] , g(x) is not equal to zero

so numeraor

given condition is

f(x) | g(x) true ,

if F(x) may have no zero divisors , then

we have deg [f(x) g(x) ] = deg f(x) + deg(g(x)

here

f(x) = a0 +a1x+................+an x^n

g(x) = b0 +b1x+ .......... +bm x^m

given g(x) can\'t be zero

and f(X) might be zero

so deg f(x) <= deg g(x)