1. Polynomial Functions and Equations

In this section, I propose that we use Scientific Notebook (or some other Computer Algebra System) instead of using the (quite useless) Factor and Remainder Theorems. (I have reluctantly included them in the next section, Remainder and Factor Theorems).


Definition of a Polynomial

A polynomial function of degree n is of the form:

f(x) = a0xn + a1xn-1 + a2xn-2 + ... + an

where

a0 ≠ 0 and

n is a positive integer, called the degree of the polynomial.

Example:

f(x) = x4 − x3 − 19x2 − 11x + 31 is a polynomial function of degree 4.


Roots of an Equation

Finding the roots of an equation, for example

x4 − x3 − 19x2 − 11x + 31 = 0,

means to find values of x which make the equation true. Here we use Scientific Notebook to solve such equations.

Solution 1: Graphically. The roots are given by the x-intercepts.

f(x) = x4 − x3 − 19x2 − 11x + 31
1_Polynomials__10.png

We see that there are 4 roots, at approximately

x = -3, x = -2, x = 1, x = 5.


Solution 2: Using "Compute → Solve → Numeric" in Scientific Notebook:

f(x) = x4 − x3 − 19x2 − 11x + 31,

Solution is: {x = -2.97}, {x = -2.05}, {x = 1.02}, {x = 4.99}.

We see that the roots are close to our estimation from the graph.

dinosaur

So you still insist on seeing the dinosaur method of solving these? Okay, here you go...

In the next section we meet the Remainder Theorem and Factor Theorem, which were historically used to find solutions of polynomial equations. They are only useful for polynomial equations with simple roots.



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