Hotmath
Math Homework. Do It Faster, Learn It Better.

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 3 x 2 + 21 x + 18 can be factored as

3 x 2 + 21 x + 18 = 3 ( x + 1 ) ( x + 6 ) .

The polynomial 5 x + 10 can be factored as

5 x + 10 = 5 ( x + 2 ) .

3 and 5 are relatively prime, and none of the binomial factors are shared. So, the two polynomials

3 x 2 + 21 x + 18 and 5 x + 10

are relatively prime.

Example 4:

The polynomial x 2 3 x 4 can be factored as

x 2 3 x 4 = ( x + 1 ) ( x 4 ) .

The polynomial 3 x 2 + 21 x + 18 can be factored as

3 x 2 + 21 x + 18 = 3 ( x + 1 ) ( x + 6 ) .

The two polynomials share a binomial factor:
( x + 1 ) .

So

x 2 3 x 4 and 3 x 2 + 21 x + 18

are not relatively prime.