Polynomial algebra
Prerequisites
By this stage you should already be familiar with expressions of the form:
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
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
Algebra is the manipulation of expressions like polynomials
expression. The manipulation of polynomials is called algebra. What follows is a summary of techniques used in algebra.
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.
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.
Multiplication and division of algebraic terms
These follow the same rules as those for arithmetic. Recall that the multiplication of a positive by a negative number gives a negative number and that the multiplication of two negative numbers gives a positive number. Algebraic terms follow the same rules as the following examples illustrate.
These follow the same rules as those for arithmetic. Recall that the multiplication of a positive by a negative number gives a negative number and that the multiplication of two negative numbers gives a positive number. Algebraic terms follow the same rules as the following examples illustrate.
Addition of algebraic terms
Consider the expression
Consider the expression
Cancellation of terms
In arithmetic we can cancel out common factors.
In arithmetic we can cancel out common factors.
We can perform the same operation of cancelling down with algebraic terms.
Expanding brackets
When two algebraic terms are to be added first before being multiplied by another term, then we place those terms in a bracket.
When two algebraic terms are to be added first before being multiplied by another term, then we place those terms in a bracket.
Expanding brackets means to remove the brackets by multiplying all of the first by all of the second. To do this, the first term in the first bracket must be multiplied by each of the two terms in the second bracket, and likewise the second term in the first bracket must be multiplied by each of the two terms in the second bracket.
Observe how as a result of this process we have the expression in the result, which are like terms. These can be collected, so the whole operation of expanding the terms is:
Factorisation
In the previous example we expanded and simplified a pair of polynomials in brackets
In the previous example we expanded and simplified a pair of polynomials in brackets
To go the other way and introduce brackets into a polynomial expression is called factorisation.
Algebraic fractions
Algebraic fractions are multiplied, divided, added and subtracted according to the same rules that govern arithmetic. Consider the following examples.
