Relatively Prime Numbers and Polynomials

Two numbers are said to be relatively prime if their greatest common factor ( GCF ) is Math formula .

Example 1:

The factors of Math formula are .

The only common factor is Math formula . So, the GCF is .

Therefore, Math formula are relatively prime.

Example 2:

The factors of Math formula are .

The greatest common factor here is Math formula .

Therefore, Math formula 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 Math formula .

Example 3:

The polynomial Math formula can be factored as

Math formula .

The polynomial Math formula can be factored as

Math formula .

Math formula are relatively prime, and none of the binomial factors are shared. So, the two polynomials

Math formula

are relatively prime.

Example 4:

The polynomial Math formula can be factored as

Math formula .

The polynomial Math formula can be factored as

Math formula .

The two polynomials share a binomial factor:
Math formula .

Math formula

are not relatively prime.

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