Relatively Prime Numbers and Polynomials
Two numbers are said to be relatively prime if their greatest common factor (GCF) is 1.
Example 1:
The factors of 20 are 1, 2, 4, 5, 10 and 20.
The factors of 33 are 1, 3, 11, and 33.
The only common factor is 1. So, the GCF is 1.
Therefore, 20 and 33 are relatively prime.
Example 2:
The factors of 45 are 1, 3, 5, 9, 15, and 45.
The factors of 51 are 1, 3, 17, and 51.
The greatest common factor here is 3.
Therefore, 45 and 51 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 1.
Example 3:
The polynomial 3x2 + 21x + 18 can be factored as
3x2 + 21x + 18 = 3(x + 1)(x + 6).
The polynomial 5x + 10 can be factored as
5x + 10 = 5(x + 2).
3 and 5 are relatively prime, and none of the binomial factors are shared. So, the two polynomials
3x2 + 21x + 18 and 5x + 10
are relatively prime.
Example 4:
The polynomial x2 + 3x – 4 can be factored as
x2 + 3x – 4 = (x + 1)(x – 4).
The polynomial 3x2 + 21x + 18 can be factored as
3x2 + 21x + 18 = 3(x + 1)(x + 6).
The two polynomials share a binomial factor:
(x + 1).
So
x2 + 3x – 4 and 3x2 + 21x + 18
are not relatively prime.
