To find the greatest common factor (GCF) for the expressions [tex]\( 15v^3 \)[/tex] and [tex]\( 12v^2 \)[/tex], follow these steps:
1. Identify the coefficients and the variables with their powers for the two expressions:
- The coefficient of the first expression is 15, and the power of [tex]\( v \)[/tex] is 3.
- The coefficient of the second expression is 12, and the power of [tex]\( v \)[/tex] is 2.
2. Find the greatest common divisor (GCD) of the coefficients 15 and 12:
- The factors of 15 are [tex]\( 1, 3, 5, \)[/tex] and [tex]\( 15 \)[/tex].
- The factors of 12 are [tex]\( 1, 2, 3, 4, 6, \)[/tex] and [tex]\( 12 \)[/tex].
- The greatest common factor of 15 and 12 is [tex]\( 3 \)[/tex].
3. Determine the smallest power of [tex]\( v \)[/tex] that appears in both expressions:
- The powers of [tex]\( v \)[/tex] are 3 and 2.
- The smallest power is [tex]\( v^2 \)[/tex].
Thus, the greatest common factor of the expressions [tex]\( 15v^3 \)[/tex] and [tex]\( 12v^2 \)[/tex] is [tex]\( 3v^2 \)[/tex].
So, the answer is:
[tex]\(3v^2\)[/tex]
