Factorize $3 x^{2} - (3 \sqrt{3} - 1) x - \sqrt{3}$ and verify relationship between the zeroes and the coefficients. [2 MARKS]

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Solution

Factorisation : 1 Mark
Verification : 1 Mark

Given polynomial is,

$3 x^{2} - (3 \sqrt{3} - 1) x - \sqrt{3}$

$= 3 x^{2} - 3 \sqrt{3} x + x - \sqrt{3}$

$= 3 x (x - \sqrt{3}) + 1 (x - \sqrt{3})$

$= (x - \sqrt{3}) (3 x + 1)$

So the zeroes are,

$x - \sqrt{3} = 0 \Rightarrow x = \sqrt{3}$

$3 x + 1 = 0 \Rightarrow x = - \frac{1}{3}$

For verification,

$α + β = \frac{- 1}{3} + \sqrt{3} = \frac{3 \sqrt{3} - 1}{3} = - \frac{b}{a}$

$α β = \frac{- 1}{3} \times \sqrt{3} = \frac{- 1}{\sqrt{3}} = \frac{c}{a}$

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