Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Definitions from Complex polynomials

All definitions from Complex polynomials.

Back to Complex polynomials

Complex polynomial

A complex polynomial is a function p:C→Cp:\mathbb C\to\mathbb C of the form

p(z)=a0+a1z+a2z2+⋯+anzn,a0,…,an∈C,p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n, \qquad a_0,\ldots,a_n\in\mathbb C,

for a non-negative integer nn.

The coefficients may be real or non-real. The word polynomial restricts the exponents of the variable to non-negative integers, not the kinds of numbers allowed as coefficients.

  1. [1]:Complex number
  2. [2]:Degree
  3. [3]:Classify and find degrees
  4. [4]:Polynomial root
  5. [5]:Polynomial factor
  6. 6 more

Degree

For a non-zero polynomial

p(z)=∑k=0nakzk,p(z)=\sum_{k=0}^n a_kz^k,

the degree of pp is the largest integer kk for which ak≠0a_k\ne0. The zero polynomial has undefined degree.

  1. [1]:Complex polynomial
  2. [2]:Classify and find degrees
  3. [3]:Roots of a high-degree complex polynomial

Polynomial root

A number α∈C\alpha\in\mathbb C is a root of pp if

p(α)=0.p(\alpha)=0.
  1. [1]:Complex polynomial
  2. [2]:Find and use roots
  3. [3]:Root multiplicity
  4. [4]:Remainder theorem
  5. [5]:Factor theorem
  6. 1 more

Polynomial factor

A polynomial p1p_1 is a factor of pp if there is a polynomial p2p_2 such that

p(z)=p1(z)p2(z)p(z)=p_1(z)p_2(z)

for every z∈Cz\in\mathbb C.

  1. [1]:Complex polynomial
  2. [2]:Find and use roots
  3. [3]:Root multiplicity
  4. [4]:Factor theorem

Root multiplicity

For a non-zero polynomial pp, a root α\alpha has multiplicity k≥1k\ge1 if (z−α)k(z-\alpha)^k is a factor of pp but (z−α)k+1(z-\alpha)^{k+1} is not.

  1. [1]:Polynomial root
  2. [2]:Polynomial factor
  3. [3]:Multiplicity two

On this page

Complex polynomialDegreePolynomial rootPolynomial factorRoot multiplicity