Question

12. If and o- are the two imaginary cube roots of unity, then the equation whose roots are aw317 and aw382 is :
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Answer 1

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=> x² + ax + a² = 0.

If w and w²are the two imaginary cube, roots of unity, then

1 + w + w² = 0

=> w + w² = -1 ....(1)

The sum of root

= aw^317 + aw^382

= a(w^317 + w^382)

= a(w + w²) = -a [from (1)]

The product of roots

= aw^317 + aw^382 = a²s^699 = a²

Therefore, the required equation is

x² - (sum of roots)x + (products of roots) = 0

=> x² + ax + a² = 0.

Note: cube roots of -1 and -1, -w, -w².

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Answer 2

Answer:

=> x² + ax + a² = 0.

If w and w²are the two imaginary cube, roots of unity, then

1 + w + w² = 0

=> w + w² = -1 ....(1)

The sum of root

= aw^317 + aw^382

= a(w^317 + w^382)

= a(w + w²) = -a [from (1)]

The product of roots

= aw^317 + aw^382 = a²s^699 = a²

Therefore, the required equation is

x² - (sum of roots)x + (products of roots) = 0

=> x² + ax + a² = 0.

Note: cube roots of -1 and -1, -w, -w².