To find the greatest common factor (GCF), you need to compare the factor trees for each set of numbers or expressions.
For the first set of numbers (40 and 128):
- The factor tree for 40 is: 2^3 * 5
- The factor tree for 128 is: 2^7
To find the GCF, you need to look for the common factors between the two numbers. In this case, the only common factor is 2 raised to the power of 3, which is 8. So, the GCF for 40 and 128 is 8, not 2.
For the second set of numbers (90, 105, and 180):
- The factor tree for 90 is: 3^2 * 2 * 5
- The factor tree for 105 is: 5 * 3 * 7
- The factor tree for 180 is: 3^2 * 2^2 * 5
To find the GCF, you need to look for the common factors between all three numbers. In this case, the common factors are only 3 and 5. The highest power of 3 is 3^2, and there is no common factor of 2 raised to the same power in all three numbers. Therefore, the GCF for 90, 105, and 180 is 3 * 5, which is equal to 15, not 5.
For the third set of expressions (3x^3, 6x, and 12x^2):
- The expression 3x^3 cannot be factored further. It represents a monomial with the variables 3 and x raised to the power of 3.
- The expression 6x is already fully factored.
- The factor tree for 12x^2 is: 2^2 * 3 * x^2
To find the GCF, you need to look for the common factors among all three expressions. In this case, there is a common factor of 3 and one power of x. Therefore, the GCF for 3x^3, 6x, and 12x^2 is 3x, not just 3.
So, in conclusion, the correct GCFs are:
- For the first set (40 and 128): GCF = 8
- For the second set (90, 105, and 180): GCF = 15
- For the third set (3x^3, 6x, and 12x^2): GCF = 3x