Abstract Algebra: Theory and Applications · Problem 17.5.4

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-gcd

Problem

Find the greatest common divisor of each of the following pairs

p(x)

and

q(x)

of polynomials. If

d(x) = \gcd(p(x), q(x))

, find two polynomials

a(x)

and

b(x)

such that

a(x)p(x) + b(x)q(x) = d(x)

. ((a))

p(x) = x^3 - 6x^2 + 14x - 15

and

q(x) = x^3 - 8x^2 + 21x - 18

, where

p(x), q(x) \in \mathbb{Q}[x]

((b))

p(x) = x^3 + x^2 - x + 1

and

q(x) = x^3 + x - 1

, where

p(x), q(x) \in \mathbb{Z}_2[x]

((c))

p(x) = x^3 + x^2 - 4x + 4

and

q(x) = x^3 + 3x - 2

, where

p(x), q(x) \in \mathbb{Z}_5[x]

((d))

p(x) = x^3 - 2x + 4

and

q(x) = 4x^3 + x + 3

, where

p(x), q(x) \in \mathbb{Q}[x]