Question

Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively 0, 5Csqrt%7B5%7D%20" id="TexFormula1" title=" \sqrt{5} " alt=" \sqrt{5} " align="absmiddle" class="latex-formula">

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

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


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