Chapter Contents

• Polynomial Equations of Higher Degree
• 1. Polynomial Functions and Equations
• 2. Remainder and Factor Theorems
• 3. Factors and Roots of Polynomial Equations
• Polynomial Equations of Higher Degree Problem Solver

• Comments, Questions?

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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 + a₁x^n-1 + a₂x^n-2 + ... + a_n

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.

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

Answer