Advanced: Relationships In Cubic Polynomials

Let us verify the relationship between the zeroes ($4$, $-2$, and $\frac{1}{2}$) and the coefficients of the cubic polynomial $p(x) = 2x^3 - 5x^2 - 14x + 8$. First, the sum of the zeroes is $4 + (-2) + \frac{1}{2} = \frac{5}{2}$, which exactly matches $-\frac{b}{a}$. Next, we calculate the sum of the products of the zeroes taken two at a time: ( $\times -2$) + ($-2 \times \frac{1}{2}$) + ($\frac{1}{2} \times 4$). This simplifies to $-8 - 1 + 2 =$ , verifying the relationship $\frac{c}{a} = \frac{-14}{2}$. Finally, we compute the product of the zeroes as $4 \times -2 \times \frac{1}{2} =$ , which corresponds to $-\frac{d}{a} = -\frac{8}{2}$.