Question

If $ {{z}^{2}}-z+1=0 $ , then find the value of $ {{z}^{n}}-{{z}^{-n}} $ , where n is a multiple of 3?
(a) $ 2{{\left( -1 \right)}^{n}} $
(b) 0
(c) $ {{\left( -1 \right)}^{n+1}} $
(d) None of these

Answer 1

Hint:
We start solving the problem by making use of the fact that $ {{z}^{3}}+1=\left( z+1 \right)\left( {{z}^{2}}-z+1 \right) $ for finding the value of $ {{z}^{3}} $ . We then use this value to find the value of $ {{z}^{n}} $ and $ {{z}^{-n}} $ for odd and even powers of n. We then substitute the obtained values in the equation $ {{z}^{n}}-{{z}^{-n}} $ and make the necessary calculations to get the required answer.

Complete step by step answer:
According to the problem, we are given that $ {{z}^{2}}-z+1=0 $ . We need to find the value of $ {{z}^{n}}-{{z}^{-n}} $ if n is a multiple of 3.
We are given that $ {{z}^{2}}-z+1=0 $ .
$\Rightarrow$ We know that $ {{z}^{3}}+1=\left( z+1 \right)\left( {{z}^{2}}-z+1 \right)=0 $ ---(1).
From equation (1), we can see that $ {{z}^{2}}-z+1 $ is a factor of $ {{z}^{3}}+1 $ . This means that z is one of the cube roots of unity.
So, we get $ {{z}^{3}}=-1 $ .
Since n is a multiple of 3, we get $ {{z}^{n}}=-1 $ and $ {{z}^{-n}}=\dfrac{1}{-1}=-1 $ for odd values. Also $ {{z}^{n}}=1 $ and $ {{z}^{-n}}=\dfrac{1}{1}=1 $ for even values.
Now, consider n is an odd multiple of 3.
$\Rightarrow$ We get $ {{z}^{n}}-{{z}^{-n}}=\left( -1 \right)-\left( -1 \right)=-1+1=0 $ ---(2).
Now, consider n is an even multiple of 3.
$\Rightarrow$ We get $ {{z}^{n}}-{{z}^{-n}}=\left( 1 \right)-\left( 1 \right)=1-1=0 $ ---(3).
From equations (2) and (3), we can see that the value of $ {{z}^{n}}-{{z}^{-n}} $ is 0.
 $ \therefore $ The correct option for the given problem is (b).

Note:
Whenever we get this type of problem, we first try to find the value of the cube of the complex number to proceed through the problem. We can also solve this problem by finding the roots of $ {{z}^{2}}-z+1=0 $ using the fact that the roots of the quadratic equation $ a{{x}^{2}}+bx+c=0 $ is $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ and then finding the value of its cubes to get the required answer. Similarly, we can expect problems to find the value of $ {{z}^{45}}+{{z}^{111}}-{{z}^{501}} $ .