Polynomial definition
A polynomial function is any function that can be written as
where n is a whole number (0, 1, 2, …), the coefficients an, …, a0 are real numbers, and an ≠ 0.
The parts
| Term | Meaning | In 4x³ − 2x² + 7 |
|---|---|---|
| Term | A number times the variable to a whole-number power | 4x³, −2x², 7 |
| Coefficient | The number multiplying the variable | 4 and −2 |
| Constant term | The term with no variable (x0) | 7 |
| Degree | The largest exponent | 3 |
| Leading term | The term with the largest exponent | 4x³ |
| Leading coefficient | The coefficient of the leading term | 4 |
By number of terms: a monomial has one term, a binomial two and a trinomial three.
Is it a polynomial?
| Expression | Polynomial? | Why |
|---|---|---|
| 5x12 − 2x6 + x5 − 198x + 1 | Yes | Whole-number exponents; degree 12 |
| 5x − 7 | Yes | Degree 1 |
| −8 | Yes | A constant, degree 0 (−8x0) |
| ∛5 x4 − (7/12)x² | Yes | Roots and fractions are fine in the coefficients |
| 4x6 + 15x−8 + 1 | No | Negative exponent |
| 5√x − x + x² | No | √x is x1/2, a fractional exponent |
| 2/x + x³ − 2 | No | Variable in a denominator (2x−1) |
| 2x | No | Variable in the exponent |
Simplify before deciding: 2x³ · 3x + 4 = 6x4 + 4 is a polynomial, and so is −x(x² − 4) = −x³ + 4x.
Types by degree
| Degree | Name | Example | Graph |
|---|---|---|---|
| 0 | Constant | f(x) = 5 | Horizontal line |
| 1 | Linear | f(x) = 2x + 1 | Straight line |
| 2 | Quadratic | f(x) = −2x² + 4x − 1 | Parabola |
| 3 | Cubic | f(x) = x³ − 3x² − 4x | S-shaped curve |
End behavior
For large |x|, the leading term dominates, so it alone decides what the graph does at the far left and right.
| Degree | Leading coefficient | As x → −∞ | As x → ∞ |
|---|---|---|---|
| Even | Positive | f(x) → ∞ | f(x) → ∞ |
| Even | Negative | f(x) → −∞ | f(x) → −∞ |
| Odd | Positive | f(x) → −∞ | f(x) → ∞ |
| Odd | Negative | f(x) → ∞ | f(x) → −∞ |
Even degree: both ends point the same way. Odd degree: the ends point opposite ways.
Zeros and turning points
- A polynomial of degree n has at most n − 1 turning points.
- The multiplicities of its zeros add up to at most n, so it has at most n x-intercepts.
- At a zero with even multiplicity the graph touches the x-axis and turns around; at a zero with odd multiplicity it crosses.
Worked example: f(x) = x³ − 3x² − 4x
- Degree and leading coefficient: the leading term is x³, so the degree is 3 and the leading coefficient is 1.
- End behavior: odd degree, positive leading coefficient, so f(x) → −∞ as x → −∞ and f(x) → ∞ as x → ∞.
- Zeros: factor: x(x² − 3x − 4) = x(x − 4)(x + 1). Zeros at x = −1, 0 and 4, each with multiplicity 1, so the graph crosses the axis at each.
- Turning points: at most 3 − 1 = 2.
- Y-intercept: f(0) = 0.
Second example: f(x) = −2x² + 4x − 1 is a quadratic (degree 2) with leading coefficient −2, so both ends go down: f(x) → −∞ as x → ±∞. Its graph is a downward parabola with one turning point.
Work with polynomials
Find zeros with the polynomial roots calculator, multiply polynomials with the polynomial multiplication calculator, and see end behavior and turning points on the graphing calculator. For the slope and shape of a polynomial graph, see concavity and inflection points.
Sources: OpenStax Calculus Volume 1, 1.2 Basic Classes of Functions; OpenStax College Algebra 2e, 5.2; OpenStax Precalculus 2e, 3.4; Paul's Online Notes: Polynomials.
Every worked example on this page was checked with Calcist's own engine and an independent reference.