Question

find the greatest common factor of the expressions.
$z^2$, $-z^6$

Explanation:

Step1: Analyze the coefficients and variables

The coefficients of \( z^2 \) and \( -z^6 \) are 1 and -1 respectively. The greatest common factor of 1 and -1 is 1 (or -1, but we can consider the positive common factor for the variable part first and then account for the sign later, but in this case, the variable part will determine the GCF). For the variable part, we have \( z^2 \) and \( z^6 \).

Step2: Find the GCF of the variable exponents

To find the greatest common factor of the powers of \( z \), we look at the smallest exponent among the two. The exponents are 2 and 6. The smallest exponent is 2. So the GCF of \( z^2 \) and \( z^6 \) is \( z^2 \). Now, considering the coefficients, the GCF of 1 and -1 is 1 (or we can take -1, but since \( z^2 \) is a factor of both \( z^2 \) and \( -z^6 \) (because \( -z^6 = -z^{2 + 4}= -z^2 \cdot z^4 \)), the greatest common factor of \( z^2 \) and \( -z^6 \) is \( z^2 \) (we can also note that the sign of the GCF can be positive as \( z^2 \) divides both, and the negative sign is part of the second term's coefficient which doesn't affect the variable part's GCF here).

Answer:

\( z^2 \)