Question

Which expression is equivalent to (z−3)^4/(z−6) for all values of z where the expression is defined?

A. z^2
B. z^7
C. 1/z^18
D. 1/z^6

Answer 1

The expression (z-3)^4/(z-6) is equivalent to z^4 - 12z^3 + 54z^2 - 108z + 81.

Step-by-step explanation:

The expression (z-3)^4/(z-6) can be simplified by expanding the numerator and then canceling out common factors with the denominator. Applying the binomial theorem to the numerator, we get:

(z-3)^4 = z^4 - 12z^3 + 54z^2 - 108z + 81

Now, dividing the expanded numerator by (z-6), we can cancel out (z-6) from the expanded numerator to get:

(z^4 - 12z^3 + 54z^2 - 108z + 81)/(z-6)

Therefore, the expression (z-3)^4/(z-6) is equivalent to z^4 - 12z^3 + 54z^2 - 108z + 81.