### Functions: Polynomials

### The notion of polynomial

Polynomial

A **polynomial i**s an expression of the form:

\[a_0+a_1x+a_2x^2+\cdots + a_nx^n\tiny,\]where #a_0#, #a_1#, #a_2,\ldots,a_n# are numbers (the **coefficients** of the polynomial) and #x# is the variable.

If #a_n\ne0#, then #n# is called the **degree** of the polynomial. The number #a_n# is then called the **leading coefficient** of the polynomial.

By way of convention, we say that the polynomial #0# is of degree #-1#.

The above polynomial defines a function #f# with rule

\[f(x) =a_0+a_1x+a_2x^2+\cdots + a_nx^n\tiny.\]Such a function is called a **polynomial function**.

If #f# is a polynomial function of degree at most #n#, then we can find the polynomial function rule from the values #f(x_0), f(x_1),\ldots,f(x_n)# of #f# in #n+1# different points #x_0, x_1,\ldots,x_n#.

The degree of #0# differs from the degree of each other constant. Here is an explanation: the leading coefficient of the polynomial #3# is equal to #3# because it is the coefficient of #x^0#, the highest power that occurs in the polynomial, which is non-zero. For #0# rather than #3#, that statement is not true.

The corresponding graph is a straight line at height #y=4#.

In general, finding the zeros of a polynomial, or: the solution of the equation \[a_0+a_1x+a_2x^2+\cdots + a_nx^n =0\] is very difficult. We have the *quadratic formula*, which gives the zeros of a quadratic function. For polynomial functions of degree 3 there is a similar formula, ascribed to Cardano. For polynomials of degree 4, there also exists a formula that gives all the zeros of the fourth degree polynomial in terms of the coefficients. It was demonstrated (by Abel and independently by Galois) that for fifth degree polynomials and higher, there is no such formula. (That does not mean we cannot find the zeros of such an equation, because we can. But these solutions cannot be obtained by a single formula).

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