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]

Theorems from Complex polynomials

All theorems from Complex polynomials.

Back to Complex polynomials

Remainder theorem

When p(z)p(z) is divided by z−αz-\alpha, the remainder is the complex number p(α)p(\alpha).

  1. [1]:Polynomial root
  2. [2]:Proof
  3. [3]:Factor theorem

Factor theorem

A number α∈C\alpha\in\mathbb C is a root of pp if and only if z−αz-\alpha is a factor of p(z)p(z).

  1. [1]:Polynomial root
  2. [2]:Polynomial factor
  3. [3]:Remainder theorem
  4. [4]:Proof
  5. [5]:Uniqueness of polynomial coefficients
  6. [6]:Factorisation over the complex numbers
  7. [7]:A complex conjugate pair of roots
  8. 1 more

Uniqueness of polynomial coefficients

Two complex polynomials agree for every z∈Cz\in\mathbb C if and only if their corresponding coefficients are equal. In particular, a complex polynomial is zero for every zz if and only if all its coefficients are zero.

This is why identities between polynomial functions can be checked coefficient by coefficient.

To see why, subtract the two polynomials. If the result were a non-zero polynomial of degree dd, it could have at most dd distinct roots: each root supplies a linear factor by the factor theorem. After dividing out z−αz-\alpha, any other root β≠α\beta\ne\alpha remains a root of the quotient, so we can repeat this step at most dd times. It therefore cannot vanish at every complex number.

  1. [1]:Complex polynomial
  2. [2]:Factor theorem

Fundamental theorem of algebra

Every complex polynomial of degree n≥1n\ge1 has at least one complex root.

The theorem is stated without proof. It guarantees that a root exists, not a formula for finding it.

Applying the factor theorem to that root and then repeating with the quotient, gives the full factorisation below. This is what is special about C\mathbb C: over R\mathbb R, a polynomial such as z2+1z^2+1 cannot be factored into real linear factors.

  1. [1]:Complex polynomial
  2. [2]:Factorisation over the complex numbers
  3. [3]:Roots of a real polynomial
  4. [4]:Seen questions
  5. [5]:Seen questions
  6. 1 more

Factorisation over the complex numbers

Every complex polynomial of degree n≥1n\ge1 can be written

p(z)=a(z−α1)(z−α2)⋯(z−αn),p(z)=a(z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_n),

where aa is the coefficient of znz^n and the αj∈C\alpha_j\in\mathbb C are roots, counted with multiplicity.

  1. [1]:Fundamental theorem of algebra
  2. [2]:Factor theorem
  3. [3]:Proof
  4. [4]:Count roots with multiplicity
  5. [5]:Factorise a quadratic
  6. [6]:Factorise z8−1z^8-1
  7. 5 more

On this page

Remainder theoremFactor theoremUniqueness of polynomial coefficientsFundamental theorem of algebraFactorisation over the complex numbers