Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2+px+q$, from a polynomial whose zeros are ${(\alpha +\beta )}^2$ and ${(\alpha -\beta )}^2$.

Answer 1

$f\left(x\right)=x^2+px+q$

Roots are $\alpha$ and $\beta$

$\alpha +\beta =-p$

$\alpha \beta =q$

${(\alpha +\beta )}^2=p^2$

$(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$

$p^2-4q$

New Roots are $(\alpha +\beta {)}^2$ and $(\alpha -\beta {)}^2$

Sum=$(\alpha +\beta {)}^2$ + $(\alpha -\beta {)}^2$

$=p^2+p^2-4q$

$=2(p^2-2q)$

Product=$(\alpha +\beta {)}^2$ $(\alpha -\beta {)}^2$

$=p^2(p^2-4q)$

$\therefore f\left(x\right)=K\left(x^2-2(p^2-2q)x+p^2(p^2-4q)\right)$