If $x = - \frac{1}{3}$ is a zero of a polynomial $p (x) = 27 x^{3} - {a x}^{2} - x + 3$ , then find the value of $^{'} a^{'}$

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Solution

$x = - \frac{1}{3}$ is a zero of polynomial $p (x)$
$⟹ p (- \frac{1}{3}) = 0$
$⟹ 27 (- \frac{1}{3})^{3} - a (- \frac{1}{3})^{2} - (- \frac{1}{3}) + 3 = 0$
$⟹ - \frac{27}{27} - \frac{a}{9} + \frac{1}{3} + 3 = 0$
$⟹ \frac{a}{9} = 2 + \frac{1}{3}$
$⟹ \frac{a}{9} = \frac{7}{3}$

$⟹ a = 21$

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