3 Polynomial expressions

An important group of mathematical expressions which use indices are known as polynomials . Examples of polynomials are

4 x 3 + 2 x 2 + 3 x − 7 , x 2 + x , 17 − 2 t + 7 t 4 , z − z 3

Notice that they are all constructed using non-negative whole number powers of the variable. Recall that x 0 = 1 and so the number − 7 appearing in the first expression can be thought of as − 7 x 0 . Similarly the 17 appearing in the third expression can be read as 17 t 0 .

Key Point 8
Polynomials

A polynomial expression takes the form

a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … + a n x n


where a 0 , a 1 , a 2 , a 3 , … a n are all constants called the coefficients of the polynomial. The number a 0 is also called the constant term . The highest power in a polynomial is called the degree of the polynomial.

Polynomials with low degrees have special names and subscript notation is often not needed:

Polynomial Degree Name
a x 3 + b x 2 + c x + d 3 cubic
a x 2 + b x + c 2 quadratic
a x + b 1 linear
a 0 constant
Task!

Which of the following expressions are polynomials? Give the degree of those which are.

  1. 3 x 2 + 4 x + 2 ,
  2. 1 x + 1 ,
  3. x ,
  4. 2 t + 4 ,
  5. 3 x 2 + 4 x + 2 .

Recall that a polynomial expression must contain only terms involving non-negative whole number powers of the variable.

Give your answers by ringing the correct word (yes/no) and stating the degree if it is a polynomial.

  1. yes: polynomial of degree 2, called quadratic
  2. no
  3. no
  4. yes: polynomial of degree 1, called linear
  5. (e) no
Exercises
  1. State which of the following are linear polynomials, which are quadratic polynomials, and which are constants.
    1. x ,
    2. x 2 + x + 3 ,
    3. x 2 − 1 ,
    4. 3 − x ,
    5. 7 x − 2 ,
    6. 1 2 ,
    7. 1 2 x + 3 4 ,
    8. 3 − 1 2 x 2 .
  2. State which of the following are polynomials.
    1. − α 2 − α − 1 ,
    2. x 1 ∕ 2 − 7 x 2 ,
    3. 1 x ,
    4. 19.
  3. Which of the following are polynomials ?
    1. 4 t + 17 ,
    2. 1 2 − 1 2 t ,
    3. 15 ,
    4. t 2 − 3 t + 7 ,
    5. 1 t 2 + 1 t + 7
  4. State the degree of each of the following polynomials. For those of low degree, give their name.
    1. 2 t 3 + 7 t 2 ,
    2. 7 t 7 + 14 t 3 − 2 t 2 ,
    3. 7 x + 2 ,
    4. x 2 + 3 x + 2 ,
    5. 2 − 3 x − x 2 ,
    6. 42
  1. (a), (d), (e) and (g) are linear. (b), (c) and (h) are quadratic. (f) is a constant.
    1. is a polynomial, (d) is a polynomial of degree 0.
    2. and
    3. are not polynomials.
    3. and
    4. are polynomials.
    1. 3, cubic,
    2. 7,
    3. 1, linear,
    4. 2, quadratic,
    5. 2, quadratic,
    6. 0, constant.