Question

Find a polynomial equation of the lowest degree with rational co-efficient having √3, (1 – 2i) as two of its roots.

Answer 1

When √3 is a root, – √3 will also be a root. 

Now the quadratic equation with √3, – √3 are roots is x² – (√3 – √3)x + (√3)(- √3) = 0 

(i.e) x² – 3 = 0 

When 1 – 2i is a root, 1 + 2i will be another root. 

Now the quadratic equation with roots 1 – 2i and 1 + 2i is

x² – (1 – 2i + 1 + 2i)x + (1 – 2i)(1 + 2i) = 0 

(i.e) x² – 2x + 5 = 0 

∴ The equation with roots ± √3 and 1 ± 2i is 

(x² – 3) (x² – 2x + 5) = 0 

(i.e) x⁴ – 2x³ + 5x² – 3x² + 6x – 15 = 0, 

(i.e) x⁴ – 2x³ + 2x² + 6x – 15 = 0