Maths - Polynomials

On this page we are looking a functions of one variable, usually denoted by x, in the next section we will look at functions of many variables.

A polynomial has powers of the variable x multiplied by coefficients.

For instance, as the variable is x, the powers are: x, x2, x3, x4 ...

So an example of a polynomial is:

f(x) = a0+ a1 x + a2 x2+ a3 x3+ a4 x4...

A polynomial with powers up to x2 is a quadratic, quadratics occur frequently in equations of motion, solving quadratic equations is explained here.

Polynomials of infinite length can represent functions like trig functions (sin, cos, tan) therefore we can use an infinite series to calculate a value of these functions to any accuracy required as explained here.

Polynomials where the values of an and xn are rational numbers (or complex numbers made up of rational numbers) are known as Z-equations and their solutions can be studied using Galois theory.

If the polynomial is equal to zero:

a0+ a1 x + a2 x2+ a3 x3+ a4 x4... = 0

Then we can 'solve' it, that is find values of x for which the equation is true, although if the equation has terms of degree 5 or higher then there may not be a formula for its solution (see Galois theory), that is, although it has a solution in the complex numbers it may not have a solution in complex numbers containing rationals and roots of rationals.

Any equation containing only one variable, some constants, the addition and multiplication operations can be rearranged to be in this polynomial form (as long as it does not contain the division operation or functions such as sine and exponential). Perhaps that is why polynomials are so important in ring theory? perhaps this is the ring equivalent of a free group?

An alternative form of the above equation is:

(x - r0)(x - r1)(x - r2) ... = 0

The constants r0,r1.. r2 have a negative sign so that they are solutions to the equation, that is if,
x - r0= 0 then x = r0.

So we have two forms which can be made equivalent, in the first form the multiplications are done first and the additions last:

a0+ a1 x + a2 x2+ a3 x3+ a4 x4... = 0

In the second form the additions are done first and the multiplications last:

(x - r0)(x - r1)(x - r2) ... = 0

It is easy to get from the second form to the first just by multiplying out the terms. However, as we have already mentioned, it is harder to get from the first form to the second form. This is known as solving or 'splitting' the polynomial.

Solutions to cubic and quartic.

Program

There are a number of open source programs that can solve polynomial equations. I have used Axiom, how to install Axiom here.

To get a numeric solution for a given equation we can use complexSolve as shown here:

complexSolve(3*x^2+4*x+5 = 0,1.e-10)

I have put user input in red:

(1) -> complexSolve(3*x^2+4*x+5 = 0,1.e-10)

(1)
[x= - 0.6666666666 6666666667 - 1.1055415968 003217131 %i,
x= - 0.6666666666 6666666667 + 1.1055415968 003217131 %i]
Type: List Equation Polynomial Complex Float
(2) ->

Or we can find a formula for, say, a quadratic equation using radicalSolve as shown here:

(3) -> radicalSolve(a*x^2 + b*x + c = 0,x) 
       
     
              +-----------+         +-----------+
              |          2          |          2
           - \|- 4a c + b   - b    \|- 4a c + b   - b
(3)  [x= --------------------,x= ------------------]
           2a                     2a
           Type: List Equation Expression Integer
         

Linear Functions

There are two ways to define linear functions, some definitions allow a constant offset and some don't.

f(x) = a * x + b

or

f(x) = a * x

The second definition is an example of a linear map or linear operator. We can extend this for several variables as explained on this page.

 


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