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MATH
[1251]

Propositions from Complex polynomials

All propositions from Complex polynomials.

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Conjugate roots for real coefficients

If pp has real coefficients and α\alpha is a root of multiplicity kk, then α‾\overline\alpha is also a root of multiplicity kk.

The condition on the coefficients is essential: conjugation must leave every coefficient unchanged before it can pass through the polynomial.

  1. [1]:Complex polynomial
  2. [2]:Complex conjugate
  3. [3]:Proof
  4. [4]:Real linear and quadratic factors
  5. [5]:Roots of a real polynomial
  6. [6]:Seen questions

Real linear and quadratic factors

Every non-zero polynomial with real coefficients can be written as a non-zero real constant times a product of real linear factors and real quadratic factors.

Every non-real root occurs with its conjugate. Pairing the corresponding factors gives

(z−α)(z−α‾)=z2−2Re⁡(α)z+∣α∣2,(z-\alpha)(z-\overline\alpha) =z^2-2\operatorname{Re}(\alpha)z+|\alpha|^2,

whose coefficients are real. Real roots already provide real linear factors. The quadratic factors obtained from non-real roots have negative discriminant, so they do not split further over R\mathbb R.

  1. [1]:Conjugate roots for real coefficients
  2. [2]:Factorisation over the complex numbers
  3. [3]:Factorise x8−1x^8-1 over the reals
  4. [4]:Factorisation over the real and complex numbers
  5. [5]:Seen questions

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Conjugate roots for real coefficientsReal linear and quadratic factors