Find the zeroes of the following polynomial by the factorisation method and verify the relationship between the zeroes and the coefficients of the polynomial:
$v^{2}+4 \sqrt{3} v-15$

(A) Let $f(v) = v^{2} + 4 \sqrt{3} v - 15$.
To factorise,we split the middle term: $v^{2} + (5 \sqrt{3} - \sqrt{3}) v - 15 = 0$.
$= v^{2} + 5 \sqrt{3} v - \sqrt{3} v - 15 = 0$.
$= v(v + 5 \sqrt{3}) - \sqrt{3}(v + 5 \sqrt{3}) = 0$.
$= (v + 5 \sqrt{3})(v - \sqrt{3}) = 0$.
Thus,the zeroes are $v = -5 \sqrt{3}$ and $v = \sqrt{3}$.
Verification:
Sum of zeroes $= -5 \sqrt{3} + \sqrt{3} = -4 \sqrt{3} = -\frac{\text{Coefficient of } v}{\text{Coefficient of } v^{2}}$.
Product of zeroes $= (-5 \sqrt{3})(\sqrt{3}) = -5 \times 3 = -15 = \frac{\text{Constant term}}{\text{Coefficient of } v^{2}}$.
Hence,the relationship is verified.