Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

Let $f(x)=2 x^{2}-x-6$

To find the zeroes, we put $f(x)=0$

$\Rightarrow 2 x^{2}-x-6=0$

$\Rightarrow 2 x^{2}-4 x+3 x-6=0$

$\Rightarrow 2 x(x-2)+3(x-2)=0$

$\Rightarrow(x-2)(2 x+3)=0$

$\Rightarrow(x-2)=0$ or $(2 x+3)=0$

$\Rightarrow x=2,-\frac{3}{2}$

Hence, all the zeroes of the polynomial $f(x)$ are 2 and $-\frac{3}{2}$.

Now,

$f(2)=2(2)^{2}-2-6$

$=2(4)-8$

$=8-8$

$=0$

$f\left(-\frac{3}{2}\right)=2\left(-\frac{3}{2}\right)^{2}-\left(-\frac{3}{2}\right)-6$

$=2\left(\frac{9}{4}\right)+\frac{3}{2}-6$

$=\frac{9}{2}+\frac{3}{2}-6$

$=\frac{12}{2}-6$

$=6-6$

$=0$

Hence, the relationship between the zeros and the coefficients is verified.