Question

Compate the greatest common divisor h=god(f,g) of the given polynomials f,g. What is the degree deg(h) ? f(x)=x3−2x2+2x−1,g(x)=x4−x3−x+1∈Z7​

Answer 1

To find the greatest common divisor (gcd) of two polynomials, factorize each polynomial and find their common factors. In this case, the gcd of f(x) and g(x) is (x - 1) with a degree of 1.

Step-by-step explanation:

To find the greatest common divisor (gcd) of two polynomials, we need to factorize each polynomial and find their common factors. In this case, the given polynomials are f(x) = x3 - 2x2 + 2x - 1 and g(x) = x4 - x3 - x + 1 in Z7.

Let's start by factorizing f(x):

• f(x) = x3 - 2x2 + 2x - 1
• = x3 - x2 - x2 + x + x - 1
• = x2(x - 1) - x(x - 1) + 1(x - 1)
• = (x - 1)(x2 - x + 1)

Next, let's factorize g(x):

• g(x) = x4 - x3 - x + 1
• = x3(x - 1) - 1(x - 1)
• =(x - 1)(x3 - 1)
• = (x - 1)(x - 1)(x2 + x + 1)

Now, we can see that both f(x) and g(x) have a common factor of (x - 1), so the gcd of f and g is (x - 1). The degree of (x - 1) is 1, so the degree of the gcd h is 1.