If f(x) is a polynomial of degree n with rati | vedclass

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(A) Given that $f(x)$ is a polynomial with rational coefficients.If a complex number $a+bi$ is a root,its conjugate $a-bi$ must also be a root.Thus,$1

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$5$

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(A) Given that $f(x)$ is a polynomial with rational coefficients.If a complex number $a+bi$ is a root,its conjugate $a-bi$ must also be a root.Thus,$1+2i$ and $1-2i$ are roots.If a surd $a+\sqrt{b}$ is a root,its conjugate $a-\sqrt{b}$ must also be a root.Thus,$2-\sqrt{3}$ and $2+\sqrt{3}$ are roots.Additionally,$5$ is given as a root.Therefore,the roots are $1+2i, 1-2i, 2-\sqrt{3}, 2+\sqrt{3},$ and $5$.Counting these,we have $5$ distinct roots.Since the polynomial has at least these $5$ roots,the least value of $n$ is $5$.

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3},$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

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