polynome complet definition

A polynomial is what we call any function that is defined by an equation of the form p(x) = anxn + an 1xn where an an 1 a1 a0 2 R 1 + · · · + a1x + a0 Examples The following three functions are examples of polynomials • p(x) = = ( 2)x2 + ( 2x2 ⇡x + 2p2 2) so 2p is a polynomial We could rewrite

1 6 Factorizing polynomials It is often useful to write a polynomial as a product of polynomials of lower degree For example f(x) = x3 − 2x2 − x + 2 can be factorized as (x2 − 1)(x − 2) or as (x − 1)(x + 1)(x − 2) Factorizing helps find the roots since if a product of numbers is zero then one of them must be zero

A polynomial function is a function such as a quadratic a cubic a quartic and so on involving only non-negative integer powers of x We can give a general defintion of a polynomial and define its degree 2 What is a polynomial? A polynomial of degree n is a function of the form f(x) = anxn + an−1xn−1 + + a2x2 + a1x + a0

A polynomial is a combination of terms containing numbers and variables raised to positive (or zero) whole number powers Examples of Polynomials NOT polynomials (power is a fraction) (power is negative) B Terminology 1 Degree a Term Degree: sum of powers in a term the degree is the degree is the degree is 1 b

− a) b) Q(x) − · for some polynomial Q(x) Another useful fact about zeros of polynomials is given below for a polynomial of degree 3 1 2 4 The cubic polynomial revisited If a (real) polynomial P (x) = ax3 + bx2 + cx + d where a = 0 a b c and d are real has exactly 3 real zeros α β and γ then P (x) = a(x α)(x β)(x γ) (1) − − −