Find the quadratic polynomial whose zeros are 2 and −6.
Question:

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

Solution:

Let $\alpha=2$ and $\beta=-6$

Sum of the zeroes, $(\alpha+\beta)=2+(-6)=-4$

Product of the zeroes, $\alpha \beta=2 \times(-6)=-12$

$\therefore$ Required polynomial $=x^{2}-(\alpha+\beta) x+\alpha \beta=x^{2}-(-4) x-12$

$=x^{2}+4 x-12$

Sum of the zeroes $=-4=\frac{-4}{1}=\frac{-(\text { coefficient of } x)}{\text { (coefficient of } x^{2} \text { ) }}$

Product of the zeroes $=-12=\frac{-12}{1}=\frac{\text { constant term }}{\text { coefficient of } x^{2}}$