Find the highest common factor hx of fxx53x3x22x2 and gxx43x

Find the highest common factor h(x) of f(x)=x⁵+3x³+x²+2x+2 and g(x)=x⁴+3x³+3x²+x+2 in Z₅[x]. Find polynomials r(x), s(x) in Z₅[x] such that h(x)=r(x)f(x)+s(x)g(x)

Solution

Given f(x)= x5+3x3+x2+2x+2 =5x+9+2x+2x+2=9x+11

g(x)=x4+3x3+3x2+x+2=4x+9+6+x+2=5x+17

Since there is no common factor in both the polynomials

h(x)=1

Now h(x)= r(x)f(x)+s(x)g(x)

or r(x)f(x)+s(x)g(x)=1

or r(x)= 1-s(x)g(x)/f(x)=     1 -s(x) (5x+17)/(9x+11)

or s(x)= 1-r(x)f(x)/g(x) = 1-r(x) (9x+11)/ (5x+17)