## Summary of Polynomial Theorems

There are several basic theorems concerning polynomials which make the subject consistent and whole.

Every polynomial of degree with real coefficients can be factorised into complex 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)

If is a zero of a polynomial with real coefficients then its complex conjugate is 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 degreed can 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 factor then the root is said to have multiplicity Roots of are real or complex. Complex roots occur in conjugate pairs and Every root gives rise to a factor If is a root of then is a factor and When is divided by where is a root of p(z), then the remainder is 0.

When is divided by where is not a root of then the remainder is  