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Find the common roots of x^12-1=0 and x...

Find the common roots of `x^12-1=0 and x^4+x^2+1=0`

Text Solution

Verified by Experts

The correct Answer is:
`x + pm omega^(2), pm omega`
`x^(1//2)-1=(x^(6) + 1)(x^(6)-1)=(x^(6)+1)(x^(2)-1)(x^(4)+x^(2)+1)`
Common roots are given by `x^(4) + x^(2) + 1=0`
`therefore " " x^(2)=(-1pm,isqrt(3))/(2) = omega,omega^(2) or omega^(4),omega^(2)" "(because omega^(3)=1)`
` or x = pm omega^(2), pm omega`
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