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).

This is because the Factor and Remainder Theorems only really work for "nice" polynomials (that is, ones which have integer coefficients and so are easy to guess an initial root).

It's usually best to draw a graph of the function and determine the roots from where the graph cuts the `x`-axis.

Definition of a Polynomial

A polynomial function of degree n is of the form:

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


a0 ≠ 0 and

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


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.

We'll find those roots using a computer algebra system.


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

f(x) = x4 x3 − 19x2 − 11x + 31

Graph of f(x) = x4 x3 − 19x2 − 11x + 31

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.

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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.