Carcass wrote:
x is an integer.
Quantity A |
Quantity B |
(−1)x2+(−1)x3+(−1)x4 |
(−1)x+(−1)2x+(−1)3x+(−1)4x |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Key properties: (-1)^(even integer) = 1, and (-1)^(odd integer) = -1. So, we need only consider two possible cases: x is EVEN, and x is ODD.
Case i: x is ODD If x is ODD, then x^2 is odd, x^3 is odd, and x^4 is odd.
Also, if x is ODD, then 2x is even, 3x is odd and 4x is even
So, we get:
QUANTITY A:
(−1)x2+(−1)x3+(−1)x4=(−1)+(−1)+(−1)=−3QUANTITY B:
(−1)x+(−1)2x+(−1)3x+(−1)4x=(−1)+1+(−1)+1=0In this case,
Quantity B is greater. Case ii: x is EVEN If x is even, then x^2 is even, x^3 is even, and x^4 is even.
Also, if x is EVEN, then 2x is even, 3x is even and 4x is even.
So, we get:
QUANTITY A:
(−1)x2+(−1)x3+(−1)x4=1+1+1=3QUANTITY B:
(−1)x+(−1)2x+(−1)3x+(−1)4x=1+1+1+1=4In this case,
Quantity B is greater.In both possible cases,
Quantity B is greater.Answer: B