Final answer:
The common factor for the expression 30x^4y^3 - 12x^3y with a coefficient other than 1 that contains at least one variable is 6x^3y. Factoring this out of the original expression results in 6x^3y(5x - 2).
Step-by-step explanation:
To find a common factor for the expression 30x^4y^3 − 12x^3y that includes a coefficient other than 1 and contains at least one variable, we need to first identify the greatest common divisor (GCD) of the numerical coefficients and the lowest powers of the variables that appear in both terms.
The coefficients are 30 and 12, which have a GCD of 6. For the variables, x appears to be the third power in the second term and the fourth power in the first term – so x^3 is the highest power of x that is in both terms. Similarly, y appears to be the first power in the second term and the third power in the first term, so y is the common variable factor.
Combining these, the common factor we are looking for is 6x^3y. By factoring it out, we can express the original expression as:
6x^3y(5x - 2).
This illustrates how to find a common factor that simplifies the expression while being more specific than just the number 1.