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Two complex numbers α and β are such that α + β = 2 and α4 + β4 = 272, then the polynomial whose zeros are α and β is – a)x2 – 2x – 16 = 0 b)x2 – 2x + 12 = 0 c)x2 – 2x – 8 = 0 d)None of these Correct answer is option 'C'. Can you explain this answer?
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Two complex numbers α and β are such that α + β = 2 and α4 + β4 = 272,...
Solution:
Given, αβ = 2 and α4β4 = 272
We have to find the polynomial whose zeros are α and β.
Let us assume that the polynomial is of the form x² + bx + c = 0.
Using the sum and product of roots formulae, we have:

α + β = -b, and αβ = c

From, αβ = 2, we get c = 2.
Now we need to find the value of b.
Let us express α⁴ and β⁴ in terms of α and β using the identity (αβ)² = α²β²:

α⁴ + β⁴ + 2α²β² = (α² + β²)²
α⁴ + β⁴ + 2(α + β)²α²β² = (α + β)⁴
α⁴ + β⁴ + 8α²β² = (α + β)⁴

Substituting the given values, we get:

272 + 8(2) = (-b)⁴
288 = b⁴
b = ∛288

Hence, the polynomial whose zeros are α and β is:

x² + (∛288)x + 2 = 0

Simplifying, we get:

x² - 2x - 8 = 0

Therefore, the correct answer is option C (x² - 2x - 8 = 0).
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Two complex numbers α and β are such that α + β = 2 and α4 + β4 = 272,...
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Two complex numbers α and β are such that α + β = 2 and α4 + β4 = 272, then the polynomial whose zeros are α and β is – a)x2 – 2x – 16 = 0 b)x2 – 2x + 12 = 0 c)x2 – 2x – 8 = 0 d)None of these Correct answer is option 'C'. Can you explain this answer?
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