A degree 4 polynomial P(x) with integer coefficients has zeros 4 i and 4, with 4 being a zero of multiplicity 2. Moreover, the coefficient of x^4 is 1. Find the polynomial. Note: The polynomial must be expanded so that no imaginary number i appears in the polynomial.

Question

tkhunny · Answer

There's that hint, again.

1) 4i is a zero.
2) Coefficients are REAL.

Why do they keep giving the same clue?

anonymous · Answer

$$x^4-8 x^3+32 x^2-128 x+256$$ Maybe?

tkhunny · Answer

Maybe.  We're still waiting for the realization that if COMPLEX zeros exist, the ONLY way to get REAL coefficients is the have the complex zeros appear in conjugate pairs.

Thus, if you have 4i, you must also have -4i.  This gives the additional required zeros.

anonymous · Answer

Yeah, any polynomial of grade 4 has 4 roots. One is missing, I just assumed the other root was -4i.