Question

if p,q are complex roots of unity,then proved that p^2+q^2-pq= -2

Answer 1

Answer:

p^2 + q^2 - pq = - 2.

Step-by-step-explanation:

Let,

Complex roots of unity be $\omega$ and $\omega ^2$, since complex roots of unity are square of each other.

Let p be $\omega$ and q be $\omega ^2$

According to the question : If p and q are complex roots of unity -

= > p^2 + q^2 - pq

= > ( $\omega$ ) + ( $\omega ^2$ ) - ( $\omega$ x $\omega ^2$ )

= > $\omega$ + $\omega ^2$ - $\omega ^3$

= > ( $\omega$ )[ 1 + $\omega$ - $\omega ^2$ ]

From the properties of complex numbers :

• 1 + $\omega$ = - $\omega ^2$

Therefore,

= > ( $\omega$ )[ - $\omega ^2$ - $\omega ^2$ ]

= > ( $\omega$ ) [ - 2 $\omega ^2$ ]

= > - 2 $\omega ^3$

= > - 2 x 1 { $\omega ^3 = 1$ }

= > - 2

Hence proved.