Polynomials: Definition, Degree, Evaluation and Polynomial Functions
A polynomial is an expression built from variables and numbers using only addition, subtraction, multiplication and whole-number powers, such as 3x4 − 5x2 + 2x − 7. In brief: the degree is the highest power of the variable, here 4, and the leading coefficient is 3. A polynomial function is a function f(x) = anxn + … + a1x + a0, and f(2) = 3 · 16 − 5 · 4 + 4 − 7 = 25.
Polynomials are the most common algebraic objects. They model areas, volumes, trajectories and costs, and they approximate more complicated functions. This guide explains the terminology, how to find the degree, how to evaluate a polynomial and how the polynomial functions of low degree are named.
What is a polynomial?
A polynomial in x is a sum of terms of the form a xn, where a is a number called the coefficient and n is a whole number (0, 1, 2, …). The term with no variable, a0, is the constant term. A polynomial with one term is a monomial, with two terms a binomial and with three terms a trinomial.
Examples: 7x3 (monomial), x2 − 9 (binomial), 2x2 + 5x − 3 (trinomial). A polynomial may have several variables, as in x2y − 3xy + 4, but this article concentrates on one variable.
How do you find the degree and leading coefficient?
First write the polynomial in standard form, with the terms in order of decreasing powers. The degree is the exponent of the first term, and its coefficient is the leading coefficient.
- Simplify by combining like terms.
- Order the terms by descending powers.
- Read the degree and the leading coefficient from the first term.
Example 1
5x − 2x3 + 9 + x3 simplifies to −x3 + 5x + 9. The degree is 3 and the leading coefficient is −1. The coefficient of x2 is 0, since this term is missing.
Example 2: several variables
For 4x2y3 + xy, the degree of the first term is 2 + 3 = 5 and of the second 1 + 1 = 2. The degree of the polynomial is 5.
The constant 8 is a polynomial of degree 0. The zero polynomial has no defined degree, or sometimes it is assigned −∞, depending on the textbook.
How do you evaluate a polynomial?
Replace the variable by the given number and calculate, following the order of operations: powers first, then multiplications, then additions and subtractions. Use brackets for negative numbers.
Example 3
Let p(x) = 2x3 − x2 + 4x − 5. Then p(2) = 2 · 8 − 4 + 8 − 5 = 15. And p(−1) = 2(−1) − 1 + 4(−1) − 5 = −2 − 1 − 4 − 5 = −12. And p(0) = −5, the constant term.
Example 4: Horner form
The same polynomial can be written as ((2x − 1)x + 4)x − 5. For x = 2: (2 · 2 − 1) = 3, then 3 · 2 + 4 = 10, then 10 · 2 − 5 = 15 ✓. This nested form needs fewer multiplications and is the basis of Horner’s method.
What is a polynomial function?
A polynomial function is a function defined by a polynomial: f(x) = anxn + an−1xn−1 + … + a1x + a0, with an ≠ 0. Its domain is all real numbers, and its graph is a smooth, unbroken curve with no gaps, corners or asymptotes. The value f(0) is the y-intercept and equals a0.
The simplest examples are the constant function (degree 0), the linear function (degree 1) and the quadratic function (degree 2).
Names of polynomials by degree
| Degree | Name | Example | Maximum number of real zeros |
|---|---|---|---|
| 0 | Constant | f(x) = 5 | 0 (unless f = 0) |
| 1 | Linear | f(x) = 2x − 3 | 1 |
| 2 | Quadratic | f(x) = x2 − 4 | 2 |
| 3 | Cubic | f(x) = x3 − x | 3 |
| 4 | Quartic | f(x) = x4 − 5x2 + 4 | 4 |
| 5 | Quintic | f(x) = x5 − x | 5 |
Which expressions are not polynomials?
- 1/x or x−2, since the exponent is negative.
- √x or x1/2, because the exponent is not a whole number.
- 2x, because the variable is in the exponent.
- |x| and sin x, which are not built from the allowed operations.
Where do polynomials come from?
Polynomial equations, especially quadratic ones, were solved by Babylonian mathematicians. The solution of cubic and quartic equations by radicals was found in sixteenth-century Italy by Cardano, Tartaglia and Ferrari, while Abel and Galois in the nineteenth century showed that general equations of degree five and higher cannot be solved this way. The word “polynomial” combines the Greek poly (many) with the Latin-style ending of “binomial”.
Where are polynomials used today?
The volume of a box with sides x, x + 2 and x + 5 is the polynomial x3 + 7x2 + 10x. The height of a thrown ball is a quadratic polynomial in time. Engineers approximate complex curves with polynomials, and computers use them to calculate values of functions such as sine. Splines for fonts and animation are polynomial curves.
Which mistakes are common?
Reading the degree from the first term without ordering
In 5x + 2x3, the degree is 3, not 1. Look for the highest power.
Forgetting missing terms
x3 + 1 has a coefficient 0 for x2 and for x. This is important in division.
Sign errors when evaluating
(−2)2 = 4, but −22 = −4. Use brackets around negative inputs.
Calling any expression a polynomial
Variables must not appear in a denominator, under a root or in an exponent.