Find the zeroes of the following polynomial $5 x^{2} - 4 - 8 x$ and verify the relationship between zeroes of the polynomials and the coefficients of the polynomial.

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Solution

Given polynomial is $5 x^{2} - 8 x - 4$
Here, $a = 5, b = - 8$ and $c = - 4$

$5 x^{2} - 8 x - 4 = 5 x^{2} - 10 x + 2 x - 4$
$= 5 x (x - 2) + 2 (x - 2)$
$= (x - 2) (5 x + 2)$

So, the value of $5 x^{2} - 8 x - 4$ is zero when $x = 2$ or $x = \frac{- 2}{5}$

Threrfore , the zeroes of $5 x^{2} - 8 - 4$ are $2$ and $\frac{- 2}{5}$ .

Now, sum of zeroes $= 2 + (\frac{- 2}{5}) = \frac{8}{5} = \frac{- b}{a}$

product of zeroes $= 2 \times (\frac{- 2}{5}) = \frac{- 4}{5} = \frac{c}{a}$

Hence verified.

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