Question

find the zeroes of polynomial f(x)=x³+5x²-16x+80,if its two zeroes are equal in magnitude but opposite in sign

Answer 1

let the zeros of polynomial
$\alpha \: \: \: ( - \alpha ) \: \: and \ \: \: \: gamma \\ sum \: of \: zeros \: \alpha - \alpha + \gamma = \frac{ - b}{a} = - 5 \\ \gamma = - 5 \\ product \: of \: zeros \: - { \alpha }^{2} \gamma = \frac{ - d}{a} \\ = - 80 \\ - { \alpha }^{2} ( - 5) = - 80 \\ { \alpha }^{2} = \frac{ - 80}{5} = - 1 6 \\ \alpha = \sqrt{ - 16} \\ \alpha = + - 4i \\ so \: zeros \: of \: polynomials \: are \: - 5 \: \: \: - 4i \: \: \: \: \: + 4i$