Bunuel wrote:

For any numbers w and z, \(w#z = w^3z(8 - w^2)\). If both z and w#z are positive numbers, which of the following could be a value of w?

A. 9

B. 3

C. 0

D. −2

E. −9

We have to go through the answer choices. Remember, we want w#z to be positive.

A. 9

If w = 9, then 9#z = (9^3)z(8 - 9^2) is negative, since 9^3 and z are positive, but 8 - 9^2 is negative.

B. 3

If w = 3, then 3#z = (3^3)z(8 - 3^3) is negative, since 3^3 and z are positive, but 8 - 3^2 is negative.

C. 0

If w = 0, then 0#z = (0^3)z(8 - 0^2) is zero, since 0^3 = 0.

D. -2

If w = -2, then -2#z = ((-2)^3)z(8 - (-2)^2) is negative, since z and 8 - (-2)^2 are positive, but (-2)^3 is negative.

E. -9

If w = -9, then -9#z = ((-9)^3)z(8 - (-9)^2) is positive, since z is positive, but (-9)^3 and 8 - (-9)^2 are negative. Notice that negative x positive x negative is positive.

Answer: E

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