What is a polynomial?
A polynomial is a sum of terms, each a number times a variable raised to a whole-number power.
| Expression | Polynomial? | Why |
|---|---|---|
| 3x⁴ − 2x + 7 | yes | whole-number exponents |
| 5 | yes | a constant |
| x⁻² + 1 | no | negative exponent |
| √x + 3 | no | exponent ½ |
| 4/x | no | x in a denominator |
Naming by number of terms
| Terms | Name | Example |
|---|---|---|
| 1 | monomial | 7x³ |
| 2 | binomial | x² − 9 |
| 3 | trinomial | 2x² + 5x − 3 |
| 4 or more | polynomial | x³ + x² − x + 1 |
Naming by degree
The degree is the highest exponent.
| Degree | Name | Example |
|---|---|---|
| 0 | constant | 8 |
| 1 | linear | 3x + 2 |
| 2 | quadratic | x² − 4x + 1 |
| 3 | cubic | 2x³ − x |
| 4 | quartic | x⁴ − 16 |
| 5 | quintic | x⁵ + x |
For several variables, the degree of a term is the sum of its exponents: 3x²y³ has degree 5.
Standard form and key parts
Write terms in descending order of degree: standard form.
In −2x³ + 5x² − x + 9:
- Degree: 3
- Leading coefficient: −2
- Constant term: 9
Polynomial functions
A polynomial function looks like f(x) = aₙxⁿ + … + a₁x + a₀. Their graphs are smooth and continuous, with no breaks or sharp corners.
- Degree 1: a line
- Degree 2: a parabola
- Higher degrees: curves with more turns (at most n − 1 turning points)
End behaviour
The degree and leading coefficient tell how the graph behaves far to the left and right:
| Degree | Leading coefficient | Left end | Right end |
|---|---|---|---|
| even | positive | up | up |
| even | negative | down | down |
| odd | positive | down | up |
| odd | negative | up | down |
Operations
- Add and subtract by combining like terms.
- Multiply by distributing (degrees add).
- Divide using long or synthetic division.
Worked examples
Example 1: Classify 4x² − 1. Quadratic binomial.
Example 2: Degree and leading coefficient of 7 − 3x + x⁴? Standard form x⁴ − 3x + 7 → degree 4, leading coefficient 1.
Example 3: End behaviour of f(x) = −x³ + 2x? Odd degree, negative → up on the left, down on the right.
Common mistakes
- Reading the leading coefficient before writing in standard form.
- Calling expressions with negative or fractional exponents polynomials.
- Confusing the number of terms with the degree.
Try it yourself
- Is 2x^(1/2) + 1 a polynomial?
- Classify 6x³ by terms and degree.
- Degree of 3x²y⁴ − xy?
- End behaviour of f(x) = 2x⁴ − x?
Answers: 1) no 2) cubic monomial 3) 6 4) up on both ends
Quick recap
- A polynomial has whole-number exponents.
- Name it by terms (monomial, binomial, trinomial) and by degree.
- The leading coefficient and degree determine end behaviour.