If fx is a polynomial of degree n with rational coefficients and 1+2i, 2-3 and 5 are three roots of fx=0, then the least value of n is

TS EAMCET · Maths · Quadratic Equation

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

A $5$
B $4$
C $3$
D $6$

Verified Solution

Answer & Solution

Correct Answer

(A) $5$

Step-by-step Solution

Detailed explanation

Given that f(x) is a polynomial, with rational coefficients. Also, 1+2i, 2-3, 5 are roots of f(x). Since, imaginary and surd roots always exist in pairs when the coefficients are rational, hence 1-2i, 2+3 are also roots of f(x)=0 ∴ f(x) has 5 at least…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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