Question

if alpha and beta are the zeroes of f(x)=x^2+px +q form a polynomial whose zeroes are( alpha+beta)^2 and (alpha -beta)^2

Answer 1

${x}^{2} + px + q \\ \alpha + \beta = - p \: \: \: \: \: eq1\\ \alpha \beta = q \\ now \: find \: the \: value \: of \: {( \alpha + \beta) }^{2} \: \: \: \: \: from \: eq1 \\ {( \alpha + \beta) }^{2} = {p}^{2} \\ {( \alpha - \beta) }^{2} = ( { \alpha + \beta )}^{2} - 4 \alpha \beta \\ = {p}^{2} - 4q \\ now \: sum \: of \: new \: zeros \: i.e \\ { \alpha + \beta }^{2} + ( { \alpha - \beta) }^{2} = {p }^{2} + {p}^{2} - 4q \\ = 2 {p}^{2} - 4q = \frac{ - b}{a} \\product \: of \: zeros \: {( \alpha + \beta )}^{2} ( { \alpha - \beta )}^{2} \\ = {p}^{2}( {p}^{2} - 4q) \\ = \frac{c}{a} \\ polynomial \: is \\ {x}^{2} + \frac{b}{a} x + \frac{c}{a} \\ {x}^{2} - (2 {p}^{2} - 4q)x + {p}^{2} ( {p}^{2} - 4q)$