Factorize $z^{2} + 3 \sqrt{3} z - 12$ and verify the relationship between zeroes and coefficient [2 MARKS]

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Solution

Factorisation : 1 Mark
Verification : 1 Mark

Given polynomial is,

$z^{2} + 3 \sqrt{3} z - 12$

$= z^{2} + 4 \sqrt{3} z - \sqrt{3} z - 12$

$= (z + 4 \sqrt{3})$ $(z - \sqrt{3})$

So the zeroes are,

$z + 4 \sqrt{3} = 0 \Rightarrow z = - 4 \sqrt{3}$

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

For verification,

$α + β = - 4 \sqrt{3} + \sqrt{3} = - 3 \sqrt{3} = - \frac{b}{a}$

$α β = - 12 = \frac{c}{a}$

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