Let g(x) be a cubic polynomial with leading coefficient unity such that the roots of g(x)=0 are the squares of the roots of x3+x+1=0, then which of the following is/are TRUE?

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Let g(x) be a cubic polynomial with leading coefficient unity such that the roots of $g (x) = 0$ are the squares of the roots of $x^{3} + x + 1 = 0,$ then which of the following is/are TRUE?

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g(3) is a prime number

last two digit of ${(g (1))}^{50}$ are 49

units digit of ${(g (2))}^{2023}$ is 3

g(x)=0 has 3 distinct real roots

answer is A, B, C.

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Detailed Solution

$α^{2} = x \Rightarrow α = \sqrt{x}$ replace x by $\sqrt{x}$ in $x^{3} + x + 1 = 0$ . We get $g (x) = x^{3} + 2 x^{2} + x - 1$

$\Rightarrow g (3) = 47 \Rightarrow g (1) = 3 \Rightarrow 3^{50} = {(10 - 1)}^{25} = {}^{25}C_{0} 10 {}^{25}+ ..... {}^{25}C_{23} 10 {}^{2}+ {}^{25}C_{24} 10 - 1$