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Question:

Find all zeroes of the polynomial 2x⁴ - 9x³ + 5x² + 3x - 1 if two of its zeroes are 2 + √3 and 2 - √3.

Solution:

Given polynomial p(x) = 2x⁴ - 9x³ + 5x² + 3x - 1
Zeroes of the polynomial p(x) are 2 + √3 and 2 - √3
Let α, β be the other two zeroes.
We know that,
Sum of the roots = 2 + √3 + 2 - √3 + α + β = 9/2
=> 4 + α + β = 9/2
=> α + β = 1/2 (1)
Product of the roots = (2 + √3)(2 - √3) × α × β = -1/2
=> (4 - 3) × α × β = -1/2
=> α × β = -1/2 (2)
On solving equation (1) and (2) we get
α = 1 and β = -1/2