1 Polynomial functions

A very important type of function is the polynomial . Polynomial functions are made up of multiples of non-negative whole number powers of a variable, such as 3 x 2 , − 7 x 3 and so on. You are already familiar with many such functions. Other examples include:

P 0 ( t ) = 6

P 1 ( t ) = 3 t + 9 (The linear function you have already met).

P 2 ( x ) = 3 x 2 − x + 2

P 4 ( z ) = 7 z 4 + z 2 − 1

where t , x and z are independent variables.

Note that fractional and negative powers of the independent variable are not allowed so that f ( x ) = x − 1 and g ( x ) = x 3 ∕ 2 are not polynomials. The function P 0 ( t ) = 6 is a polynomial - we can regard it as 6 t 0 .

By convention a polynomial is written with the powers either increasing or decreasing. For example the polynomial

3 x + 9 x 2 − x 3 + 2

would be written as

− x 3 + 9 x 2 + 3 x + 2  or  2 + 3 x + 9 x 2 − x 3

In general we have the following definition:

Key Point 11

A polynomial expression has the form

a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 2 x 2 + a 1 x + a 0

where n is a non-negative integer, a n , a n − 1 , … , a 1 , a 0 are constants and x is a variable.

A polynomial function P ( x ) has the form

P ( x ) = a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 2 x 2 + a 1 x + a 0

The degree of a polynomial or polynomial function is the value of the highest power. Referring to the examples listed above, polynomial P 2 has degree 2, because the term with the highest power is 3 x 2 , P 4 has degree 4, P 1 has degree 1 and P 0 has degree 0. Polynomials with low degrees have special names given in Table 5.

Table 5

degree name
a 0 constant
a x + b 1 linear
a x 2 + b x + c 2 quadratic
a x 3 + b x 2 + c x + d 3 cubic
a x 4 + b x 3 + c x 2 + d x + e 4 quartic

Typical graphs of some polynomial functions are shown in Figure 30. In particular, observe that the graphs of the linear polynomials, P 1 and Q 2 are straight lines.

Figure 30 :

{ Graphs of some typical linear, quadratic and cubic polynomials}

Task!

Which of the polynomial graphs in Figure 30 are odd and which are even? Are any periodic ?

P 2 is even. P 3 is odd. None are periodic.

Task!

State which of the following are polynomial functions. For those that are, state the degree and name.

  1. f ( x ) = 6 x 2 + 7 x 3 − 2 x 4
  2. f ( t ) = t 3 − 3 t 2 + 7
  3. g ( x ) = 1 x 2 + 3 x
  4. f ( x ) = 16
  5. g ( x ) = 1 6
  1. polynomial of degree 4 (quartic),
  2. polynomial of degree 3 (cubic),
  3. not a polynomial,
  4. polynomial of degree 0 (constant),
  5. polynomial of degree 0 (constant)
Exercises
  1. Write down a polynomial of degree 3 with independent variable t .
  2. Write down a function which is not a polynomial.
  3. Explain why y = 1 + x + x 1 ∕ 2 is not a polynomial.
  4. State the degree of the following polynomials:
    1. P ( t ) = t 4 + 7 ,
    2. P ( t ) = − t 3 + 3 ,
    3. P ( t ) = 11 ,
    4. P ( t ) = t
  5. Write down a polynomial of degree 0 with independent variable z .
  6. Referring to Figure 27, state which functions are one-to-one and which are many-to-one.
  1. For example f ( t ) = 1 + t + 3 t 2 − t 3 .
  2. For example y = 1 x .
  3. A term such as x 1 ∕ 2 , with a fractional index, is not allowed in a polynomial.
    1. 4,
    2. 3,
    3. 0,
    4. 1.
  5. P ( z ) = 13 , for example.
  6. P 1 , Q 1 and P 3 are one-to-one. The rest are many-to-one.