Question

Find the quadratic polynomial, the sum of whose zeros is −5 and their product is 6. Hence, find the zeros of the polynomial.

Answer 1

$\begin{array}{l}\text{Let}\alpha \text{and}\beta \text{be the zeros of the required polynomial}f\left(x\right)\text{.}\\ \text{Then (}\alpha \text{+}\beta \text{)=}-5\text{and}\alpha \beta \text{=6}\\ \therefore f\left(x\right)=x^2-\text{(}\alpha \text{+}\beta \text{)}x+\alpha \beta \\ =>f\left(x\right)=x^2-(-5)x+6\\ =>f\left(x\right)=x^2+5x+6\\ \text{Hence, the required polynomial is}f\left(x\right)=x^2+5x+6.\\ \therefore f\left(x\right)=0=>x^2+5x+6=0\\ \text{=>}{x}^2+3x+2x+6=0\\ \text{=>}x(x+3)+2(x+3)=0\\ \text{=> (}x\text{+2)}(x+3)=0\\ \text{=> (}x\text{+2)}=0\text{or}(x+3)=0\\ \text{=>}x=-2\text{or}x=-3\\ \text{So, the zeros of}f\left(x\right)\text{are}-2\text{and}-3.\end{array}$