Question

find the polynomial whose zeros are 3+√5 and 3-√5

Answer 1

Dear Student,

Standard Quadratic polynomial is
$a {x}^{2} + bx + c$



sum of zeros
$\alpha + \beta = \frac{ - b}{a}$
$\frac{ - b}{a} = 3 + \sqrt{5} + 3 - \sqrt{5} \\ = 6$
Product of zeros
$\alpha \beta = \frac{c}{a} \\ \frac{c}{a} = (3 + \sqrt{5} )(3 - \sqrt{5} ) \\ = 9 - 5 \\ = 4$
polynomial is
${x}^{2} - ( \frac{ - b}{a} )x + \frac{c}{a} = 0 \\ {x}^{2} - ( 6 )x + 4 = 0 \\ \\ {x}^{2} - 6x + 4$
is that polynomial whose zeros are 3+√5 and 3-√5.