What is the highest common factor of \(12x^3\) and \(18x^2\)?

To determine the highest common factor (HCF) of (12x^3) and (18x^2), we begin by breaking down each coefficient and variable into their respective factors.

The coefficient of (12x^3) is 12, which can be factored into (2^2 \cdot 3). The coefficient of (18x^2) is 18, which can be factored into (2 \cdot 3^2).

Next, we identify the greatest common factor of the coefficients. For the factors:

• From (12) (which is (2^2 \cdot 3)), we have:

• (2^0) (not present) or (2^1) (the minimum power) contributes (2^1).

• (3^1) (taking the minimum power of (3)) contributes (3^1).

Combining these gives the HCF of the coefficients: (2^1 \cdot 3^1 = 2 \cdot 3 = 6).

Now, looking at the variable parts, we take the lowest power of (x