1 Multiplying polynomials together
Key Point 7
A polynomial expression is one of the form
[maths rendering] must be a positive integer.
For example [maths rendering] is a polynomial expression in [maths rendering] . The polynomial may be expressed in terms of a variable other than [maths rendering] . So, the following are also polynomial expressions:
[maths rendering]
Note that only non-negative whole number powers of the variable (usually [maths rendering] ) 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 [maths rendering] has degree 3, [maths rendering] has degree 6, and [maths rendering] has degree 1.
Let us consider what happens when two polynomials are multiplied together. For example
[maths rendering]
is the product of two first degree polynomials. Expanding the brackets we obtain
[maths rendering]
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
[maths rendering]
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 [maths rendering] , state the degree of the undefined polynomial.
second.
Task!
- If [maths rendering] , state the degree of the undefined polynomial.
- What is the coefficient of [maths rendering] in this unknown polynomial ?
- First.
- It must be 3 in order to generate the term [maths rendering] when the brackets are removed.
Task!
If [maths rendering] (a polynomial), what must be the coefficient of [maths rendering] in this unknown polynomial ?
It must be 2 in order to generate the term 2 [maths rendering] when the brackets are removed.
Task!
Two quadratic polynomials are multiplied together. What is the degree of the resulting polynomial?
Fourth degree.