§4.6 Exercises

Exercises 4.6.A

Find all the roots in \(\mathbb{C}\) of each polynomial (one root is already given):

(a) \(x^4 - 3x^3 + x^2 + 7x - 30\); root: \(1 - 2i\)

(b) \(x^4 - 2x^3 - x^2 + 6x - 6\); root: \(1 + i\)

Find a polynomial in \(\mathbb{R}[x]\) that satisfies the given conditions:

(a) Monic of degree 3 with 2 and \(3 + i\) as roots

(b) Monic of least possible degree with \(1 - i\) and 2 as roots

Factor each polynomial as a product of irreducible polynomials in \(\mathbb{Q}[x]\), in \(\mathbb{R}[x]\), and in \(\mathbb{C}[x]\):

(a) \(x^4 - 2\)

(b) \(x^3 + 1\)

Factor \(x^2 + x + 1 + i\) in \(\mathbb{C}[x]\).

Show that a polynomial of odd degree in \(\mathbb{R}[x]\) with no multiple roots must have an odd number of real roots.

Let \(f(x) = ax^2 + bx + c \in \mathbb{R}[x]\) with \(a \neq 0\). Prove that the roots of \(f(x)\) in \(\mathbb{C}\) are

\[ \frac{-b + \sqrt{b^2 - 4ac}}{2a} \quad \text{and} \quad \frac{-b - \sqrt{b^2 - 4ac}}{2a}. \]

Hint: Show that \(ax^2 + bx + c = 0\) is equivalent to \(x^2 + (b/a)x = -c/a\); then complete the square to find \(x\).

Prove that every \(ax^2 + bx + c \in \mathbb{R}[x]\) with \(b^2 - 4ac < 0\) is irreducible in \(\mathbb{R}[x]\). Hint: See Exercise 6.

If \(a + bi\) is a root of \(x^3 - 3x^2 + 2ix + i - 1 \in \mathbb{C}[x]\), then is it true that \(a - bi\) is also a root?