Find a polynomial equation of minimum degree with rational coefficient

1.	Find a polynomial equation of minimum degree with rational coefficients, having √5 – √3 as a root.
Answer» The given one roots of the polynomial equation are (√5 – √3) The other roots are (√5 + √3), (-√5 + √3) and (- √5 – √3). The quadratic factor with roots (√5 – √3) and (√5 + √3) is = x² – x(S.O.R) + P.O.R = x² – x(2√5) + (√5 – √3) (√5 + √3) = x² – 2√5 x + 2 The other quadratic factors with roots (-√5 + √3) (-√5 – √3) is = x²– x (S.O.R) + P.O.R = x² – x (-2√5 ) + (5 – 3) = x² + 2√5x + 2 To rationalize the co-efficients with minimum degree (x² – 2√5 x + 2) (x² + 2√5 x + 2) = 0 ⇒ (x² + 2)² – (2√5 x)² = 0 ⇒ x⁴ + 4 + 4x² – 20x² = 0 ⇒ x⁴ – 16x² + 4 = 0

Find a polynomial equation of minimum degree with rational coefficients, having √5 – √3 as a root.

Answer»

The given one roots of the polynomial equation are (√5 – √3)

The other roots are (√5 + √3), (-√5 + √3) and (- √5 – √3).

The quadratic factor with roots (√5 – √3) and (√5 + √3) is

= x² – x(S.O.R) + P.O.R

= x² – x(2√5) + (√5 – √3) (√5 + √3)

= x² – 2√5 x + 2

The other quadratic factors with roots (-√5 + √3) (-√5 – √3) is

= x²– x (S.O.R) + P.O.R

= x² – x (-2√5 ) + (5 – 3)

= x² + 2√5x + 2

To rationalize the co-efficients with minimum degree

(x² – 2√5 x + 2) (x² + 2√5 x + 2) = 0

⇒ (x² + 2)² – (2√5 x)² = 0

⇒ x⁴ + 4 + 4x² – 20x² = 0

⇒ x⁴ – 16x² + 4 = 0