Find the remaining solutions to the polynomial

Question

Given 3i is a solution to f(x)=x^4+2x^3+x^2+18x-72

Given 6i+1 is a solution to f(x)=2x^4-9x^3+72x^2-161x-444

Answer 1

The pertinent idea here is that when the coefficients of a polynomial are real numbers (as they are here), the roots come in conjugate pairs. So for the first polynomial, the fact that 3i is a root tells you that its conjugate, -3i, is also a root.

This means you can use polynomial long division to factor out (x-3i)(x+3i), or x²+9. These polynomials have degree 4, so when you divide them by a degree 2 polynomial, you get another degree 2 (since 4-2=2). This you can solve with the quadratic formula to get the last two roots.