Polynomials - Definition - Notation - Terminology

(introduction to polynomial functions)


In this section we introduce polynomial functions. In particular we learn about key definitions, notation and terminology that should be used and understood when working with polynomials. In particular we learn about each of the following:

  • the leading term and leading coefficient
  • the degree of a polynomial

Polynomial Functions

A polynomial functions is sum of one or more powers of \(x\): \[f(x) = ax^n + bx^{n-1} + \dots + rx + s\] where \(n\) is an non-negative integer, \(n\geq 0\). A more general, and perhaps better, way of writing this is: \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0\]

For example, the functions: \[\begin{aligned} f(x) & = 5 \\ f(x) & = 2x - 3 \\ f(x) & = 3x^2 + 2x - 1 \\ f(x) & = 0.5x^3 +2x^2 + 10 \\ f(x) & = -3x^4 - 4x^2 + 7x -3 \\ f(x) & = 0.1x^5 + 2x^4 - 3x^3 + x - 1 \\ & \vdots \end{aligned}\] are all polynomial functions, often referred to as polynomials.

Remember: these functions are only polynomial functions if the powers of \(x\) are greater than, or equal to, \(0\). They are not polynomial functions if any of the powers of \(x\) are negative or fractions.


Degree, Leading Term & Coefficients of a Polynomial

Here we learn some important definitions, that must be known, when working with polynomials.

Degree of a Polynomial

Given a polynomial function, its degree is equal to the highest power of \(x\) in the polynomial.

For instance, given the polynomial: \[f(x) = 5x^6 - 4x^3 + 8x^2 + 9\] its degree is \(6\), since the highest power of \(x\) is \(x^6\).

Leading Term

The leading term of a polynomial is term which has the highest power of \(x\).

For instance, given the polynomial: \[f(x) = 6x^8 + 5x^4 + x^3 - 3x^2 - 3\] its leading term is \(6x^8\), since it is the term with the highest power of \(x\).

Coefficients

Given a polynomial function, its coefficients are the numbers multiplying each power of \(x\)

.

If a power of \(x\) is "missing", we say that its coefficient is \(0\).

The polynomial function: \[f(x) = x^4 - 2x^3 + 5x^2 + 7x + 10\] has coefficients: \(1, \ -2, \ 5, \ 7\) and \(10\).

The polynomial function: \[f(x) = -2x^5 + 4x^3 - 6x +10\] has coefficients: \(-2, \ 0, \ 4, \ 0, \ -6\) and \(10\).

Leading Coefficient

Given a polynomial function, its leading coefficient is the leading term's \(x\) coefficient.

For example, the polynomial function: \[f(x) = 6x^5 - 7x^4+3x^2 + x -10\] has leading coefficient \(6\), since the leading term is \(6x^5\) and its coefficient is \(6\).

Exercise 1

For each of the following polynomials:

  1. state the degree, \(deg\begin{bmatrix}f(x)\end{bmatrix}\).
  2. state the leading term and the leading coefficient
  3. list the coefficients
  1. \(f(x) = -3x^5 + x^4 - 2x^3 + 6x^2 + 2x - 8\)
  2. \(f(x) = 6x^7 + 3x^5 + 2x^4 - 4x + 1\)
  3. \(f(x) = -x^2+3x - 9\)
  4. \(f(x) = \frac{x^5}{2} - 3x^4 + 6x^2 - 4\)
  5. \(f(x) = 2x^2 - 5x^3\)

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Solution Without Working

  1. \(f(x) = -3x^5 + x^4 - 2x^3 + 6x^2 + 2x - 8\)
    1. The degree is \(deg\begin{bmatrix}f(x)\end{bmatrix} = 5\).
    2. The leading term is \(-3x^5\), the leading coefficient is \(a_5 = -3\).
    3. This polynomial's coefficients are: \(a_5 = -3\), \(a_4 = 1\), \(a_3 = -2\) , \(a_2 = 6\), \(a_1 = 2\) and \(a_0 = -8\).
  2. \(f(x) = 6x^7 + 3x^5 + 2x^4 - 4x + 1\)
    1. The degree is \(deg\begin{bmatrix}f(x)\end{bmatrix} = 7\).
    2. The leading term is \(6x^7\), the leading coefficient is \(a_7 = 6\).
    3. This polynomial's coefficients are: \(a_7 = 6\), \(a_6 = 0\), \(a_5 = 3\), \(a_4 = 2\), \(a_3 = 0\), \(a_2 = 0\), \(a_1 = -4\) and \(a_0 = 1\).
  3. \(f(x) = -x^2+3x - 9\)
    1. The degree is \(deg\begin{bmatrix}f(x)\end{bmatrix} = 2\).
    2. The leading term is \(-x^2\), the leading coefficient is \(a_2 = -1\).
    3. This polynomial's coefficients are: \(a_2 = -1, \ a_1 = 3\) and \(a_0 = -9\).
  4. \(f(x) = \frac{x^5}{2} - 3x^4 + 6x^2 - 4\)
    1. The degree is \(deg\begin{bmatrix}f(x)\end{bmatrix} = 5\).
    2. The leading term is \(\frac{x^5}{2}\), the leading coefficient is \(a_5 = \frac{1}{2}\).
    3. This polynomial's coefficients are: \(a_5 = \frac{1}{2}\), \(a_4 = -3\), \(a_3 = 0\), \(a_2 = 6\), \(a_1 = 0\) and \(a_0 = -4\).
  5. \(f(x) = 2x^2 - 5x^3\)
    1. The degree is \(deg\begin{bmatrix}f(x)\end{bmatrix} = 3\).
    2. The leading term is \(-5x^3\), the leading coefficient is \(a_3 = -5\).
    3. This polynomial's coefficients are: \(a_3 = -5\), \(a_2 = 2\), \(a_1 = 0\) and \(a_0 = 0\).


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