Question

Compute The Greatest Common Divisor H=Gcd(F,G) Of The Given Polynomials F,G. What Is The Degree Deg(H) ?

f(x)=x⁶ −1,g(x)=x⁵ −x⁴ +x³ +x ² −x+1∈Z ³ [x].

Answer 1

To find the greatest common divisor (GCD) of the given polynomials, factorize both polynomials and find their common factors. The GCD of the polynomials f(x) and g(x) is (x - 1)(x + 1)(x² + 1) and its degree is 3.

Step-by-step explanation:

To find the greatest common divisor (GCD) of the given polynomials, f(x) = x⁶ - 1 and g(x) = x⁵ - x⁴ + x³ + x² - x + 1, we need to factor both polynomials and find their common factors.

Factorizing f(x), we can write it as (x³ + 1)(x³ - 1). Factorizing g(x), we get g(x) = (x - 1)(x + 1)(x² + 1)(x² - x + 1).

Since the GCD is the product of the common factors raised to their lowest powers, the GCD of f(x) and g(x) is (x - 1)(x + 1)(x² + 1).

The degree of the GCD, H, is the highest power of x in the GCD expression. In this case, the degree of H is 3.