Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros for 3, 5+ square root of 5

Answer 1

The general form of a polynomial is: x² + bx + c = 0. Substituting the roots,

3² + b(3) + c = 0
3b + c = -9 --> eqn 1

(5 + √5)² + (5 +√5)(b) + c = 0
(5+√5)b + c = -30-10√5 --> eqn 2

Solving both equations simultaneously by subtracting eqn 2 from eqn 1,
(-2 - √5)b = 21 + 10√5
b = -8 - √5

Using eqn 1,
3(-8 - √5) + c = -9
-24 - 3√5 + c = -9
c = -9 + 24 + 3√5
c = 15 + 3√5

Hence, the polynomial is: f(x) = x² + (-8 - √5)x + (15 + 3√5).