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Find the zeroes of the polynomial $p(x) = 2x^2 + 5x + 2$ and verify the relationship between zeroes of $p(x)$ and its coefficients.
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Solution: $p(x) = 2x^2 + 5x + 2 = (2x + 1)(x + 2)$
Zeroes are $-\frac{1}{2}, -2$
Sum of zeroes $= -\frac{1}{2} + (-2) = -\frac{5}{2} = \frac{-\text{coefficient of } x}{\text{coefficient of } x^2}$
Product of zeroes $= -\frac{1}{2} \times (-2) = \frac{2}{2} = \frac{\text{constant term}}{\text{coefficient of } x^2}$
Zeroes are $-\frac{1}{2}, -2$
Sum of zeroes $= -\frac{1}{2} + (-2) = -\frac{5}{2} = \frac{-\text{coefficient of } x}{\text{coefficient of } x^2}$
Product of zeroes $= -\frac{1}{2} \times (-2) = \frac{2}{2} = \frac{\text{constant term}}{\text{coefficient of } x^2}$