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.Since the coefficients are rational,imaginary roots must occur in conjugate pairs.Giv

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$5$

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(A) Given that $f(x)$ is a polynomial with rational coefficients.Since the coefficients are rational,imaginary roots must occur in conjugate pairs.Given root $1+2i$ implies $1-2i$ is also a root.Similarly,irrational roots (surds) must occur in conjugate pairs.Given root $2-\sqrt{3}$ implies $2+\sqrt{3}$ is also a root.The root $5$ is a rational number and does not require a conjugate pair.Therefore,the roots of $f(x)=0$ are $1+2i, 1-2i, 2-\sqrt{3}, 2+\sqrt{3}$,and $5$.Counting these,we have a total of $5$ roots.Thus,the least degree $n$ of the polynomial $f(x)$ 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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