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Find the zeroes of the polynomial $f(t) = t^2 + 4\sqrt{3}t - 15$ and verify the relationship between the zeroes and the coefficients of the polynomial.
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$$\begin{aligned}& f(t) = t^2 + 4\sqrt{3}t - 15 \\ & = t^2 + 5\sqrt{3}t - \sqrt{3}t - 15 \\ & = (t - \sqrt{3}) (t + 5\sqrt{3})\ \therefore\end{aligned}$$ Zeroes of given polynomial are $$\begin{aligned}& -5\sqrt{3}, \sqrt{3} \\ & Sum of the zeroes =\end{aligned}$$(-5√3 + √3) = -4√3 = -coefficient of t/coefficient of t²
Product of the zeroes = $(-5\sqrt{3}) \times \sqrt{3}) = -15 = \frac{\text{constant term}}{\text{coefficient of t}^2}$
Product of the zeroes = $(-5\sqrt{3}) \times \sqrt{3}) = -15 = \frac{\text{constant term}}{\text{coefficient of t}^2}$