Construct a polynomial of the fourth degree whose coefficients are rational numbers,one of whose roots is 2i [i = square root of (-1)] and which has two irrational roots.

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- May 5th 2010, 03:54 AMaravindv01Polynomial Generation
Construct a polynomial of the fourth degree whose coefficients are rational numbers,one of whose roots is 2i [i = square root of (-1)] and which has two irrational roots.

- May 5th 2010, 04:34 AMSwlabr
In a rational polynomial, complex roots always come in pairs of conjugates (because of the formula $\displaystyle \frac{-b \pm \sqrt{b^2-4ac}}{2a}$).

So, as 2i is a root its complex conjugate is also a root. What is the complex conjugate of 2i?

Now, there is a rational polynomial of the form $\displaystyle ax^2+bx+c$ with these two complex numbers as roots. What is it?* This will then give you your answer...

*A hint for this bit would be that if p and q are roots of a quadratic polynomial P then P=(x-p)(x-q). Then expand.