Polynomial Algebra

Prerequisites

By this stage you should already be familiar with expressions of the form (x + 2) (x2 − x − 1) and x3 + x2 − 3x − 2 .

The purpose of this chapter is to summarise and consolidate the knowledge that you should already have acquired over some time regarding the manipulation of such expressions.

Polynomials

Expressions of the form

x3 + x2 − 3x − 2

are called polynomials. The expression x in this example is a variable. It stands for a number. Polynomials may have more than one variable.

In the following there are two variables, x and y

x2y + 2x2 + y2 + 2y .

Any letter or symbol can be used as a variable. Terms such as x2y that form a part of such

polynomials such as x2y + 2x2 + y2 + 2y are also called algebraic terms.

The manipulation of polynomials - algebra

There are a number of ways in which we can change a polynomial into another polynomial or expression. The manipulation of polynomials is called algebra. What follows is a summary of techniques used in algebra.

1.Substitution of values

A variable represents a quantity in general. At a given point in an algebraic process we may wish to substitute a particular value for a variable.

Example (1)

Substitute x = 2, and y = −1 in x2y + 2x2 + y2 + 2y .

Solution

When x = 2, and y = −1

x2y + 2x2 + y2 + 2y = (2)2 × −1 + 2× (2)2 + (−1)2 + 2× −1

=−4 + 8 + 1 − 2

=3