Question

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

Answer 1

$\begin{array}{l}\text{Let}\alpha \text{and}\beta \text{be the zeros of the required polynomial}f\left(x\right)\text{.}\\ \text{Then (}\alpha \text{+}\beta \text{)=0 and}\alpha \beta \text{=}-\text{1}\\ \therefore f\left(x\right)=x^2-\text{(}\alpha \text{+}\beta \text{)}x+\alpha \beta \\ =>f\left(x\right)=x^2-0x+(-1)\\ =>f\left(x\right)=x^2-1\\ \text{Hence, the required polynomial is}f\left(x\right)=x^2-1.\\ \therefore f\left(x\right)=0=>x^2-1=0\\ \text{=> (}x+1\text{)}(x-1)=0\\ \text{=> (}x+1\text{)}=0\text{or}(x-1)=0\\ \text{=>}x=-1\text{or}x=1\\ \text{So, the zeros of}f\left(x\right)\text{are}-1\text{and 1}.\end{array}$