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Prove that x^3+y^3+z^3-3xyz =(x+y+z)(x+...

Prove that `x^3+y^3+z^3-3xyz`
`=(x+y+z)(x+omegay+omega^2z)(x+yomega^2+z omega)`

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`R.H.S.=(x+y+z)(x+omegay+omega^2z)(x+yomega^2+zomega)`
`=(x+y+z)(x+xyomega^2+zxomega+xyomega+y^2omega^3+yzomega^2+zxomega^2+yzomega^4+z^2omega^3)`
`=(x+y+z)[x^2+y^2+z^2+xy(w^2+w)+xy(omega^2+omega)+zx(omega^2+omega)]`
`(x+y+z)[x^2+y^2+z^2-xy-yz-zx)`
`=x^3+y^3+z^3-3xyz="L.H.S.(Proved)"`
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MBD PUBLICATION-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-QUESTION BANK
  1. (2+5omega +2 omega^2)^6=(2+2omega+5 omega^2)^6=729

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  2. (1-omega+omega^2)(1-omega^2+omega^4)(1-omega^4+omega^2)….to 2n Factor...

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  3. Prove that x^3+y^3+z^3-3xyz =(x+y+z)(x+omegay+omega^2z)(x+yomega^2+z ...

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  4. If x=a+b, y=a omega+b omega^2,z=a omega^2+bomega show thatxyz =a^3+b^3

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  5. If x=a+b, y=a omega+b omega^2,z=a omega^2+bomega show that x^2+y^2+z^2...

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  6. If x=a+b, y=a omega+b omega^2,z=a omega^2+bomega show thatx^3+y^3+z^3=...

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  7. If ax+by +cz=X,cx+by +az=Y,bx+ay+cz=Z show that (a^2+b^2+c^2-ab-bc-ca...

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  8. If |z1|le1,|z2|le1"show that" |1-z1z2|^2-|z1-z2|^2=(1-|z1|^2)(1-|z2|^2...

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  9. If a, b, c are complex numbers satisfying a+b+c=0 and a^2+b^2+c^2=0 th...

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  10. What do the {z:|z-a|+|z+a|=2c} "where" |a|lt c represent?

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  11. What do the{z:|z-a|-|z+a|=c} represent?

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  12. What happen in (i)|a|ge c?

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  13. Given cos alpha+cos beta +cos gamma=sin alpha+sin beta +sin gamma=0 S...

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  14. Use Binomial theorem and De Moivre's theorem to show cos 3 theta=4 cos...

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  15. Find square root of -5+12sqrt(-1)

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  16. Find square root of -11-60 sqrt(-1)

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  17. Find square root of -47+8 sqrt(-1)

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  18. Find square root of -8+sqrt(-1)

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  19. Find square root of a^2-1+2asqrt(-1)

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  20. Find square root of 4ab -2(a^2-b^2) sqrt(-1)

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