Question

Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively 0,
`√(5)`

Answer 1

a + b = 0
ab = √(5)

since b = -a, subst into other equation

ab = √5
a(-a) = √5
-a² = √5
a² = - √5
a =± (5)^(1/4)·i

so our two zeros are (5)^(1/4) · i and -(5)^(1/4) · i.

Our polynomial has equation


`y = \left( x - 5^(1/4) \cdot \mathrm{i}\right) \left( x + 5^(1/4) \cdot \mathrm{i}\right)`

difference of squares: (x + y)(x - y) = x² - y² so


`\\ y = (x)^2 - \left( 5^(1/4) \cdot \mathrm{i} \right)^2 \\ y = x^2 - 5^(1/2) \cdot i^2 \\ y = x^2 + 5 ^(1/2) = x^2 + √(5)`


your polynomial is
y = x² + √(5)
y = x^2 + sqrt(5)