Question

Find all the zeros of 2x⁴ – 13x³ + 19x² + 7x – 3 if two of its zeros are `(2 + sqrt(3))` and `(2 - sqrt(3))`.

Answer 1

Let α = `(2 + sqrt(3))` and β = `(2 - sqrt(3))`.

Then, α + β = 4 and αβ = (4 – 3) = 1.

A polynomial whose zeros are α and β is x² – (α + β)x + αβ, i.e., x² – 4x + 1.

Now, divide 2x⁴ – 13x³ + 19x² + 7x – 3 by x² – 4x + 1 to give (2x² – 5x – 3) as quotient and 0 as remainder.

2x² – 5x – 3 = 0

⇒ 2x² – 6x + x – 3 = 0

⇒ 2x(x – 3) + (x – 3) = 0

⇒ (x – 3)(2x + 1) = 0

⇒ x – 3 = 0 or 2x + 1 = 0

⇒ x = 3 or x = `(-1)/2`