Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1.
Question:

Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1. Hence, find the zeros of the polynomial.

Solution:

Let $\alpha$ and $\beta$ be the zeros of the required polynomial $f(x)$.

Then $(\alpha+\beta)=0$ and $\alpha \beta=-1$

$\therefore f(x)=x^{2}-(\alpha+\beta) x+\alpha \beta$

$=>f(x)=x^{2}-0 x+(-1)$

$=>f(x)=x^{2}-1$

Hence, the required polynomial is $f(x)=x^{2}-1$.

$\therefore f(x)=0=>x^{2}-1=0$

$=>(x+1)(x-1)=0$

$=>(x+1)=0$ or $(x-1)=0$

$=>x=-1$ or $x=1$

So, the zeros of $f(x)$ are $-1$ and 1 .