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Use the Euclidean Algorithm to find the greatest common divisor of the polynomials f(x) = x^3 + 3x^2 + 6x + 1 and g(x) = 5x^4 + 2x^2 +x + 1 in Z7[x] .

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botellok

Answer:

3x+1.

Step-by-step explanation:

First we divide g(x)/f(x) (the process is in the first image):

5x-15 in Z7[x] is 5x-1 and [tex]17x^2+86x+16[/tex] is [tex]r(x)= 3x^2+2x+2[/tex] in Z7[x]. So

g(x)/f(x) = [tex](5x-1)(x^3+3x^2+6x+1)+3x^2+2x+2[/tex]

Now gcd(g,f) = gcm(f,r).

f(x)/r(x) = [tex]5x(3x^2+2x+2) + 3x+1[/tex]

Then, gcd(f,r) = gcd(r,3x+1).

r/(3x+1) = [tex](x+5)(3x+1) +4[/tex]

Then, gcd(r, 3x+1) = gcd(3x+1,4) = 3x+1.

So, gcd(f,g) = 3x+1.

Ver imagen botellok

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