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
[tex]y = \left( x - 5^{1/4} \cdot \mathrm{i}\right) \left( x + 5^{1/4} \cdot \mathrm{i}\right) [/tex]
difference of squares: (x + y)(x - y) = x² - y² so
[tex]\\ 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 + \sqrt{5}[/tex]
your polynomial is
y = x² + √(5)
y = x^2 + sqrt(5)
