Questions: Find the greatest common factor of these two expressions. 28 v^3 x^8 and 16 v^7 y^2 x^5

Transcript text: Find the greatest common factor of these two expressions. \[ 28 v^{3} x^{8} \text { and } 16 v^{7} y^{2} x^{5} \]

Solution

Solution Steps

To find the greatest common factor (GCF) of two multivariate monomials, we need to determine the GCF of the coefficients and the lowest power of each variable that appears in both monomials.

Identify the coefficients of the two monomials and find their GCF.
For each variable, find the lowest power that appears in both monomials.
Combine the GCF of the coefficients with the lowest powers of each variable.

Step 1: Find the GCF of the Coefficients

The coefficients of the two monomials are \( 28 \) and \( 16 \). The greatest common factor (GCF) of these coefficients is calculated as follows: \[ \text{GCF}(28, 16) = 4 \]

Step 2: Determine the Lowest Powers of Each Variable

Next, we analyze the variables in both monomials:

For \( v \): The powers are \( 3 \) in the first monomial and \( 7 \) in the second monomial. The lowest power is \( \min(3, 7) = 3 \).
For \( x \): The powers are \( 8 \) in the first monomial and \( 5 \) in the second monomial. The lowest power is \( \min(8, 5) = 5 \).
For \( y \): The powers are \( 0 \) in the first monomial and \( 2 \) in the second monomial. The lowest power is \( \min(0, 2) = 0 \).

Step 3: Construct the GCF Monomial

Combining the GCF of the coefficients with the lowest powers of each variable, we have: \[ \text{GCF} = 4 v^3 x^5 \]