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 $1,2,4,5,10,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20$ .

The factors of $33$ are $1,3,11,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}33$ .

The only common factor is $1$ . So, the GCF is $1$ .

Therefore, $20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}33$ are relatively prime.

Example 2:

The factors of $45$ are $1,3,5,9,15,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}45$ .

The factors of $51$ are $1,3,17,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}51$ .

The greatest common factor here is $3$ .

Therefore, $45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}51$ 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 $3{x}^{2}+21x+18$ can be factored as

$3{x}^{2}+21x+18=3\left(x+1\right)\left(x+6\right)$ .

The polynomial $5x+10$ can be factored as

$5x+10=5\left(x+2\right)$ .

$3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5$ are relatively prime, and none of the binomial factors are shared. So, the two polynomials

$3{x}^{2}+21x+18\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5x+10$

are relatively prime.

Example 4:

The polynomial ${x}^{2}-3x-4$ can be factored as

${x}^{2}-3x-4=\left(x+1\right)\left(x-4\right)$ .

The polynomial $3{x}^{2}+21x+18$ can be factored as

$3{x}^{2}+21x+18=3\left(x+1\right)\left(x+6\right)$ .

The two polynomials share a binomial factor:
$\left(x+1\right)$ .

So

${x}^{2}-3x-4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3{x}^{2}+21x+18$

are not relatively prime.