Relatively Prime Numbers and Polynomials
Two numbers are said to be relatively prime if their greatest common factor ( GCF ) is .
Example 1:
The factors of are .
The factors of are .
The only common factor is . So, the GCF is .
Therefore, are relatively prime.
Example 2:
The factors of are .
The factors of are .
The greatest common factor here is .
Therefore, are not relatively prime.
The definition can be extended to polynomials . In this case, there should be no common variable or polynomial factors, and the scalar coefficients should have a GCF of .
Example 3:
The polynomial can be factored as
.
The polynomial can be factored as
.
are relatively prime, and none of the binomial factors are shared. So, the two polynomials
are relatively prime.
Example 4:
The polynomial can be factored as
.
The polynomial can be factored as
.
The two polynomials share a binomial factor:
.
So
are not relatively prime.