Write the greatest common divisor over the given filed as a linear combination of the given...

Write the greatest common divisor over the given filed as a linear combination of the given polynomials. That is, given f(x) and g(x), find a(x) and b(x) so that d(x) = a(x)f(x) + b(x)g(x), where d(x) is the greatest common divisor of f(x) and g(x).

(a) x^10 − x^7 − x^5 + x^3 + x^2 − 1 and x^8 − x^5 − x^3 + 1 over Q.

(b) x^5 + x^4 + 2x^2 − x − 1 and x^3 + x^2 − x over Q.

(c) x^3 − 2x^2 + 3x + 1 and x^3 + 2x + 1 over Z5.

(d) x^5 + x^4 + 2x^2 + 4x + 4 and x^3 + x^2 + 4x over Z5.

I know the GCD's are

a. x^3-1 b. 1 c. 1 d.1

Please help me write them as a linear combination.

Expert Solution

Colby Messinger answered 2 years ago

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