Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

Given polynomial is

[tex]\sf \:3{x}^{2} - 2 \\ \\ [/tex]

Let assume that

[tex]\sf \: f(x) = {3x}^{2} - 2 \\ [/tex]

[tex]\sf \: f(x) = {( \sqrt{3} x)}^{2} - {( \sqrt{2}) }^{2} \\ [/tex]

[tex]\sf \: f(x) = ( \sqrt{3}x + \sqrt{2} )( \sqrt{3}x - \sqrt{2} ) \\ [/tex]

So, to find zeroes of f(x), we substitute

[tex]\sf \: f(x) = 0 \\ [/tex]

[tex]\sf \: ( \sqrt{3}x + \sqrt{2})( \sqrt{3}x - \sqrt{2}) = 0 \\ [/tex]

[tex]\implies\sf \: x = - \dfrac{ \sqrt{2} }{ \sqrt{3} } \: \: or \: \: x = \dfrac{ \sqrt{2} }{ \sqrt{3} } \\ [/tex]

Now, Consider

[tex]\sf \: Sum\:of\:zeroes \\ [/tex]

[tex]\sf \: = \: - \dfrac{ \sqrt{2} }{ \sqrt{3} }+ \dfrac{ \sqrt{2} }{ \sqrt{3} } \\ [/tex]

[tex]\sf \: = \: 0 \\ [/tex]

can be rewritten as

[tex]\sf \: = \: - \: \dfrac{0}{3} \\ [/tex]

[tex]\sf \: = \: - \: \dfrac{\: coefficient \: of \: x \: }{coefficient \: of \: {x}^{2}} \\ [/tex]

Hence,

[tex]\sf\implies \sf \: Sum\:of\:zeroes \: = \: - \: \dfrac{\: coefficient \: of \: x \: }{coefficient \: of \: {x}^{2}} \\ [/tex]

Now, Consider

[tex]\sf \: Product\:of\:zeroes \\ [/tex]

[tex]\sf \: = \: - \dfrac{ \sqrt{2} }{ \sqrt{3} } \times \dfrac{ \sqrt{2} }{ \sqrt{3} } \\ [/tex]

[tex]\sf \: = \:-\: \dfrac{2}{3} \\ [/tex]

[tex]\qquad\sf \: = \: \dfrac{constant \: term}{coefficient \: of \: {x}^{2}} \\ [/tex]

Hence,

[tex]\sf\implies \sf \:Product\:of\:zeroes \: = \: \dfrac{constant \: term}{coefficient \: of \: {x}^{2}} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information:

[tex]\bf \:If\:\alpha, \beta \: are \: zeroes \: of \: {ax}^{2} + bx + c, \: then \\ [/tex]

[tex]\sf \: \alpha + \beta = - \dfrac{b}{a} \\ [/tex]

[tex]\sf \: \alpha \beta = \dfrac{c}{a} \\ [/tex]

[tex]\bf \:If\: \alpha, \beta, \gamma \: are \: zeroes \: of \: {ax}^{3} + {bx}^{2} + cx + d, \: then\\ [/tex]

[tex]\sf \: \alpha + \beta + \gamma = - \dfrac{b}{a} \\ [/tex]

[tex]\sf \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}\\ [/tex]

[tex]\sf \: \alpha \beta \gamma = - \dfrac{d}{a} \\ [/tex]