While learning about polynomials, I came across this problem
$$x^4 + px^3 + qx^2 + rx + s = 0$$ is a polynomial with rational coefficients and $4$ complex roots. If two of the roots add up to $3 + 4i$ and the other two have a product of $13−i$, compute the value of $q$.
I let $m_1+m_2=3+4i$ and $m_3m_4=13-i$.
Since the coefficients are rational, we can apply the Complex Conjugate Roots Theorem. Because $(a+bi)(a-bi)$ always results in a real number, we know $m_3, m_4$ cannot be conjugates of each other. Hence, the conjugates are not together in the two equations.
I replaced the $m$'s with $a+bi, a-bi, c+di, c-di$ and attempted to solve for the variables. However, this led to an unwieldy system.
Is there a better way to solve this? Thanks!