greatest common divisor

(noun)

The greatest common divisor of a set is the largest positive integer or polynomial that divides each of the numbers in the set without remainder.

Related Terms

Examples of greatest common divisor in the following topics:

  • Integer Coefficients and the Rational Zeroes Theorem

    • If $a_0$ and $a_n$ are nonzero, then each rational solution $x=p/q$, where $p$ and $q$are coprime integers (i.e. their greatest common divisor is $1$), satisfies:
    • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.

  • Complex Fractions

    • Therefore, we use the cancellation method to simplify the numbers as much as possible, and then we multiply by the simplified reciprocal of the divisor, or denominator, fraction:
    • You'll find that the common denominator of the two fractions in the numerator is 6, and then you can add those two terms together to get a single fraction term in the larger fraction's numerator:

  • Simplifying, Multiplying, and Dividing

    • which, after canceling the common factor of $(x+2)$ from both the numerator and denominator, gives the simplified expression
    • Recall the rule for dividing fractions: the dividend is multiplied by the reciprocal of the divisor.

  • Introduction to Factoring Polynomials

    • Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
    • Factor out the greatest common factor, $4x(x+5) + 3y(x+5)$.