Question

Find the quadratic polynomial, the sum of whose zeros is $\left(\frac{5}{2}\right)$ and their product is 1. Hence, find the zeros of the polynomial.

Answer 1

$\begin{array}{l}\text{Let}\alpha \text{and}\beta \text{be the zeros of the required polynomial}f\left(x\right)\text{.}\\ \text{Then (}\alpha \text{+}\beta \text{)=}\frac{5}{2}\text{and}\alpha \beta \text{=1}\\ \therefore f\left(x\right)=x^2-\text{(}\alpha \text{+}\beta \text{)}x+\alpha \beta \\ =>f\left(x\right)=x^2-\frac{5}{2}x+1\\ \begin{array}{l}=>f\left(x\right)=2x^2-5x+2\\ \text{Hence, the required polynomial is}f\left(x\right)=2x^2-5x+2.\\ \therefore f\left(x\right)=0=>2x^2-5x+2=0\\ \text{=> 2}{x}^2-(4x+x)+2=0\\ \text{=> 2}{x}^2-4x-x+2=0\\ \text{=> 2}x(x-2)-1(x-2)=0\\ \text{=> (2}x-1\text{)}(x-2)=0\\ \text{=> (2}x-1\text{)}=0\text{or}(x-2)=0\\ \text{=>}x=\frac{1}{2}\text{or}x=2\\ \text{So, the zeros of}f\left(x\right)\text{are}\frac{1}{2}\text{and 2}.\end{array}\end{array}$