1 Multiplying polynomials together

Key Point 7

A polynomial expression is one of the form

a n x n + a n − 1 x n − 1 + … + a 2 x 2 + a 1 x + a 0
where a 0 , a 1 , … , a n are known coefficients (numbers), a n ≠ 0 , and x is a variable.

n must be a positive integer.

For example x 3 − 17 x 2 + 54 x − 8 is a polynomial expression in x . The polynomial may be expressed in terms of a variable other than x . So, the following are also polynomial expressions:

t 3 − t 2 + t − 3 z 5 − 1 w 4 + 10 w 2 − 12 s + 1

Note that only non-negative whole number powers of the variable (usually x ) are allowed in a polynomial expression. In this Section you will learn how to factorise simple polynomial expressions and how to solve some polynomial equations. You will also learn the technique of equating coefficients . This process is very important when we need to perform calculations involving partial fractions which will be considered in Section 6.

The degree of a polynomial is the highest power to which the variable is raised. Thus x 3 + 6 x + 2 has degree 3, t 6 − 6 t 4 + 2 t has degree 6, and 5 x + 2 has degree 1.

Let us consider what happens when two polynomials are multiplied together. For example

( x + 1 ) ( 3 x − 2 )

is the product of two first degree polynomials. Expanding the brackets we obtain

( x + 1 ) ( 3 x − 2 ) = 3 x 2 + x − 2

which is a second degree polynomial.

In general we can regard a second degree polynomial, or quadratic, as the product of two first degree polynomials, provided that the quadratic can be factorised. Similarly

( x − 1 ) ( x 2 + 3 x − 7 ) = x 3 + 2 x 2 − 10 x + 7

is a third degree, or cubic , polynomial which is thus the product of a linear polynomial and a quadratic polynomial.

In general we can regard a cubic polynomial as the product of a linear polynomial and a quadratic polynomial or the product of three linear polynomials. This fact will be important in the following Section when we come to factorise cubics.

Key Point 8

A cubic expression can always be formulated as a linear expression times a quadratic expression.

Task!

If x 3 − 17 x 2 + 54 x − 8 = ( x − 4 ) ×  (a polynomial) , state the degree of the undefined polynomial.

second.

Task!
  1. If 3 x 2 + 13 x + 4 = ( x + 4 ) ×  (a polynomial) , state the degree of the undefined polynomial.
  2. What is the coefficient of x in this unknown polynomial ?
  1. First.
  2. It must be 3 in order to generate the term 3 x 2 when the brackets are removed.
Task!

If 2 x 2 + 5 x + 2 = ( x + 2 ) × (a polynomial), what must be the coefficient of x in this unknown polynomial ?

It must be 2 in order to generate the term 2 x 2 when the brackets are removed.

Task!

Two quadratic polynomials are multiplied together. What is the degree of the resulting polynomial?

Fourth degree.