So the real part is
So your answer is correct.
Could someone, just check my answers, to see if I'm doing them right:
1) Find a real number such that:
2) Factorise into real linear and quadratic factors.
3) Find the real and imaginary parts of
Real part =
Imaginary part =
FTA claims there exists a root for p(x):
*If it is a real number a, then p(x) = (x-a)q(x).
* If the root is complex, say z, then so is the conjugate of z and thus the quadratic constructed using z and conjugate of z will then be a factor of p. But the quadratic with z and the conjugate of z as roots has real coefficients. So p(x) = (ax^2 + bx + c)q(x).
Notice that q(x) is a polynomial with real co-effs of strictly lesser degree. Now we can repeat the above steps on q(x) (and its factors) to break it into linear or quadratic factors again. Thus p(x) can be written as product of linear and quadratic polynomials.