Question

If $\alpha ,\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=x^2-2x+3$, find a polynomial whose zeroes are $\alpha +2,\beta +2$

Answer 1

$f\left(x\right)=x^2-2x+3$ have zeroes $\alpha ,\beta$

$\Rightarrow \alpha +\beta =2$

$\Rightarrow \alpha \cdot \beta =3$

Now polynomial having $\alpha +2,\beta +2$ as roots is

$p\left(x\right)=x^2-(\alpha +2+\beta +2)x+(\alpha +2)(\beta +2)$

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

$=x^2-(2+4)x+3+2\left(2\right)+4$

$\Rightarrow x^2-6x+11$