Introduction Notes Theory HOWTO Examples
Engineering Error Questions Matlab Maple
The following are some theorems about polynomials, some of which you have already seen in this class, others which may be new:
Theorem 1
A polynomial of degree n has n roots when you count multiplicity.
Definition
The complex conjugate of a complex number z = a + bj is the complex number z' = a - bj. A pair of complex numbers forms a complex conjugate pair if they are complex conjugates of each other, for example, 3 + 4j and 3 - 4j.
Theorem 2
A polynomial p(x) with real coefficients has either real roots, or complex conjugate pairs of roots.
Proof
If two roots are complex conjugate pairs z = a + bj and z' = a - bj, then (x - z)(x - z') = (x - (a + bj))(x - (a - bj)) = x2 - 2ax + a2 + b2.
Theorem 3
A polynomial p(x) with real coefficients and an odd degree has at least one real root.
Proof
There are multiple proofs, one being argued as follows: if the degree of p(x) is odd, then if p(x) → ∞ as x → ∞ then p(x) → -∞ as x → -∞ and if p(x) → -∞ as x → ∞ then p(x) → ∞ as x → -∞. Thus, in either case, the intermediate value theorem (see the bisection method) states that there must be at least one point such that p(x) = 0.
A simpler proof using Theorem 2 is as follows: assume all the roots are complex and not real. Therefore, they must come in complex conjugate pairs. Thus, the number of roots must be even. Thus, all the roots of an odd polynomial cannot be complex, and therefore one must be real.
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
