Question: Exponents and Polynomials Greatest common factor of two multivariate monomials Find the greatest…

Exponents and Polynomials

Greatest common factor of two multivariate monomials

Find the greatest common factor of these two expressions.

$$30v^2x^5 \text{ and } 18v^3x^4u^6$$

Solution

To find the greatest common factor (GCF) of two multivariate monomials, follow these steps: First, consider the coefficients and variables separately. 1. Coefficients: - The coefficients are 30 and 18. - Find the GCF of 30 and 18. - The prime factorization of 30 is \(2 \times 3 \times 5\). - The prime factorization of 18 is \(2 \times 3^2\). The common factors are \(2\) and \(3\). The GCF of the coefficients is \(2 \times 3 = 6\). 2. Variables: Consider each variable separately. - For \(v^2\) and \(v^3\): take the smallest power, which is \(v^2\). - For \(x^5\) and \(x^4\): take the smallest power, which is \(x^4\). - For \(u^0\) (since it’s not in the first expression) and \(u^6\): take the smallest power, which is \(u^0 = 1\). Now, combine the GCF of the coefficients and variables: \[ 6v^2x^4 \] So, the greatest common factor of the two expressions is \(6v^2x^4\).