If pair of polynomials have Greatest Common Factor as 1 ....

[swampwiz]

NOTE: presume real coefficients

If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a set of linear (real root) or non-decomposible quadratic factor (complex conjugate root), and so if any value is a real root of one, for it to be a root of the other, the GCF would not be 1 - and likewise for any complex number, as it would only be a root for a specific quadratic factor.

So:

factors: A( x ) , B( x )

A( c ) = 0 & B( c ) = 0 ⇒ GCF( A( c ) , B( c ) ) != 1

GCF( A( c ) , B( c ) ) = 1 ⇒ !∃ c : [ A( c ) = 0 & B( c ) = 0 ] ⇔ no common roots

!E there does not exist

Is this accurate, or am I missing something? Thanks

 

[fresh_42]

NOTE: presume real coefficients

If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a set of linear (real root) or non-decomposible quadratic factor (complex conjugate root), and so if any value is a real root of one, for it to be a root of the other, the GCF would not be 1 - and likewise for any complex number, as it would only be a root for a specific quadratic factor.

So:

factors: A( x ) , B( x )

A( c ) = 0 & B( c ) = 0 ⇒ GCF( A( c ) , B( c ) ) != 1

GCF( A( c ) , B( c ) ) = 1 ⇒ !∃ c : [ A( c ) = 0 & B( c ) = 0 ] ⇔ no common roots

!E there does not exist

Is this accurate, or am I missing something? Thanks

Usually we speak of a greatest common divisor. In addition I assume you meant $GCF(A(x),B(x))$ instead of $GCF(A(c),B(c)).$
A common zero $c \in ℝ$ of $A$ and $B$ over the reals implies a common divisor $(x-c).$
And isn't your second statement equivalent to your first, simply negated? Otherwise you should assume a non-trivial common divisor of $A$ and $B$ and handle the possibility that this could be a real number. And what if $(x^2+1)$ divides both?