Summary of Polynomial Theorems
There are several basic theorems concerning polynomials which make the subject consistent and whole.
Every polynomial of degreewith real coefficients can be factorised intocomplex linear factors. Some of these may be repeated.
Each polynomial can be factorised uniquely into the form(1)
Every real polynomial can be expressed as a product of real linear and irreducible (cannot be factoised – the coefficients may also be irrational) quadratic factors with real coefficients. (2)
Ifis a zero of a polynomial with real coefficients then its complex conjugateis also a zero.
Every polynomial of odd degree with real coeffiencets has at least one real zero. This is a consequence of (2) above. A polynomial of degreedcan be factorised into at most quadratic factors and at least 1 linear factor, all real. Each linear factor gives rise to a real root.
All real polynomials of degree n have exactly n zeros, some of which may be repeated. If the unique factorisation (1) above includes a factorthen the rootis said to have multiplicity
Roots ofare real or complex. Complex roots occur in conjugate pairsand
Every rootgives rise to a factor
Ifis a root ofthenis a factor and
Whenis divided bywhereis a root of p(z), then the remainder is 0.
Whenis divided bywhereis not a root ofthen the remainder is