In Problem, try to fill in the blanks without referring back to the text.In Problem 1 we s...

In Problem, try to fill in the blanks without referring back to the text.

In Problem 1 we saw that if z1 is a root of a polynomial equation with real coefficients, then its conjugate  is also a root. Assume that the cubic polynomial equation az3 + bz2 + cz + d = 0, where a, b, c, and d are real, has exactly three roots. One of the roots must be real because

Problem 1

If z1 is a root of a polynomial equation with real coefficients, then its conjugate  is also a root. Prove this result in the case of a quadratic equation az2 + bz + c = 0, where a ≠ 0, b, and c are real. Start with the properties of conjugates given in (1) and (2).

 (1)

 (2)