Given that \(b\) is rational and \(i=\sqrt{-1}\), the product of the expression \((3+bi)\) and which of the following expressions must be a rational number?

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If you have time, you can try out all the options.

The correct option would be to multiply \((3+bi)\) with its conjugate so we can use difference of squares:

$$ (a+b)(a-b) = a^2-b^2 $$ $$ (3+bi)(3-bi) = (3)^2-(bi)^2 $$ $$ = 9 -b^2i^2 $$ $$ = 9+b^2 $$