Relatively Prime Numbers and Polynomials
Two numbers are said to berelatively primeif theirgreatest common factor(GCF) is.
Example 1:
The factors ofare.
The factors ofare.
The only common factor is. So, the GCF is.
Therefore,are relatively prime.
Example 2:
The factors ofare.
The factors ofare.
The greatest common factor here is.
Therefore,arenotrelatively prime.
The definition can be extended topolynomials. In this case, there should be no common variable or polynomial factors, and the scalar coefficients should have a GCF of.
Example 3:
The polynomialcan be factored as
.
The polynomialcan be factored as
.
are relatively prime, and none of thebinomialfactors are shared. So, the two polynomials
are relatively prime.
Example 4:
The polynomialcan be factored as
.
The polynomialcan be factored as
.
The two polynomials share a binomial factor:
.
So
arenotrelatively prime.