Roots of Polynomials

The study of roots connects the algebraic structure of a polynomial with geometry and number theory.

Fundamental Theorem of Algebra

Theorem

Every non-constant polynomial

P (x)

with complex coefficients has at least one complex root.

Corollary: A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

P (x) = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{n})

Theorem (Rational Root Theorem)

If a polynomial with integer coefficients

a_{n} x^{n} + \dots + a_{0} = 0

has a rational root

p / q

(in lowest terms), then:

Theorem

If a polynomial has real coefficients, and

a + bi

is a root (

b \neq = 0

), then its conjugate

a - bi

is also a root.

Implication: Complex roots come in pairs. A polynomial of odd degree with real coefficients must have at least one real root.

For

P (x) = x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{0}

, all roots

z

satisfy:

∣ z ∣ < 1 + max (∣ a_{n - 1} ∣, \dots, ∣ a_{0} ∣)

Exercise (Problem 1)

Find all rational roots of

2 x^{3} - 3 x^{2} - 11 x + 6 = 0

Exercise (Problem 2)

Construct a cubic polynomial with real coefficients that has roots

2

and

1 - 3 i

Exercise (Problem 3)

Prove that

x^{4} + 3 x^{2} + 2 = 0

has no real roots.