Question

Find the quadratic polynomial whose zeros are 2 and −6. Verify the relation between the coefficients and the zeros of the polynomial.

Answer 1

$\begin{array}{l}\text{Let}\alpha \text{=2 and}\beta \text{=}-\text{6}\\ \text{Sum of the zeroes, (}\alpha +\beta )=2+(-6)=-4\\ \text{Product of the zeroes,}\alpha \beta =2\times (-6)=-12\\ \text{∴ Required polynomial =}{x}^2-\text{(}\alpha +\beta )x+\alpha \beta =x^2-(-4)x-12\\ \\ \text{=}{x}^2+4x-12\\ \text{Sum of the zeroes =}-4=\frac{-4}{1}=\frac{-\left(\text{coefficient of}x\right)}{\left(\text{coefficient of}{x}^2\right)}\\ \text{Product of the zeroes =}-12=\frac{-12}{1}=\frac{\text{constant term}}{\text{coefficient of}{x}^2}\end{array}$