What Is a Polynomial?
A polynomial is one of the most common structures in algebra — if you've ever worked with an expression like \(x^2 + 3x - 5\), you've already been working with polynomials. The word sounds technical, but the idea is simple: a polynomial is a combination of terms that follow a specific set of rules.
What Is a Polynomial?
A polynomial is an expression made up of terms that are combined using addition or subtraction, where each term is a number, a variable, or a product of numbers and variables raised to whole-number (non-negative integer) exponents.
Here are some examples:
- \(4x^3 - 2x + 7\) — a polynomial in one variable
- \(5y^2 + y\) — also a polynomial
- \(3\) — a polynomial (just a constant term)
And here are some expressions that are not polynomials:
- \(\frac{1}{x}\) — variables can't be in the denominator (that's the same as \(x^{-1}\), a negative exponent)
- \(\sqrt{x}\) — that's \(x^{1/2}\), which is not a whole-number exponent
- \(2^x\) — that's an exponential function, not a polynomial
The key rule: exponents on variables must be whole numbers (0, 1, 2, 3, …). Coefficients — the numbers out front — can be anything, including fractions and negatives.
Parts of a Polynomial
Take a close look at \(6x^4 - 3x^2 + x - 8\).
Terms are the individual pieces separated by addition or subtraction. This polynomial has four terms: \(6x^4\), \(-3x^2\), \(x\), and \(-8\).
Coefficients are the numerical factors in each term. In \(6x^4\), the coefficient is \(6\). In \(-3x^2\), it's \(-3\). The term \(x\) has an implied coefficient of \(1\).
The degree of a term is the exponent on the variable. The degree of \(6x^4\) is 4; the degree of \(-3x^2\) is 2; the degree of \(x\) is 1; the constant \(-8\) has degree 0 (since \(-8 = -8x^0\)).
The degree of the polynomial is the highest degree among all its terms. For \(6x^4 - 3x^2 + x - 8\), the degree is 4.
The leading term is the term with the highest degree, and the leading coefficient is that term's coefficient. For this polynomial, the leading term is \(6x^4\) and the leading coefficient is \(6\).
Types of Polynomials
Polynomials get classified in two ways: by the number of terms, and by degree.
By Number of Terms
| Name | Terms | Example |
|---|---|---|
| Monomial | 1 | \(7x^3\) |
| Binomial | 2 | \(x^2 - 4\) |
| Trinomial | 3 | \(x^2 + 5x + 6\) |
| Polynomial | 4 or more | \(x^3 - 2x^2 + x - 1\) |
By Degree
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | \(9\) |
| 1 | Linear | \(3x + 2\) |
| 2 | Quadratic | \(x^2 - x + 4\) |
| 3 | Cubic | \(2x^3 + x\) |
| 4 | Quartic | \(x^4 - 1\) |
| 5 | Quintic | \(x^5 + x^3\) |
You'll often combine both labels: \(3x^2 - 5\) is a quadratic binomial — degree 2, two terms.
Standard Form
A polynomial is in standard form when its terms are written from highest degree to lowest. This is just a convention — a way to organize the expression so it's easy to read at a glance.
Take \(4 + 3x^3 - x\). The degrees are 0, 3, and 1 — so rearranging from highest to lowest gives:
$$3x^3 - x + 4$$
That's standard form. The degree is 3 (cubic), and the leading coefficient is 3.
Here's another: \(5x - 2x^4 + 1 + x^2\).
Arranged in standard form:
$$-2x^4 + x^2 + 5x + 1$$
Degree 4, leading coefficient \(-2\).
One thing to watch: if a term is missing, you can still list it with a coefficient of 0. So \(-2x^4 + x^2 + 5x + 1\) could be written as \(-2x^4 + 0x^3 + x^2 + 5x + 1\) if you want to show all degree positions. This becomes useful when you're adding or subtracting polynomials and need to line up like terms.
Practice Problems
1. Which of the following are polynomials? Explain why or why not. (a) \(3x^2 - x + 5\) (b) \(x^{-2} + 4\) (c) \(\frac{2}{3}x^4 - 7\) (d) \(\frac{x+1}{x}\) Show answer(a) Yes — whole-number exponents, valid polynomial. (b) No — negative exponent. (c) Yes — fractional coefficients are fine; exponents are still whole numbers. (d) No — variable in the denominator (equivalent to \(1 + x^{-1}\)).
2. Identify the degree and leading coefficient of \(9 - 4x^3 + 2x\). Show answerFirst write in standard form: \(-4x^3 + 2x + 9\). Degree: 3. Leading coefficient: \(-4\).
3. Classify \(5x^2 - 3\) by number of terms and by degree. Show answerTwo terms → binomial. Degree 2 → quadratic. It's a quadratic binomial.
4. Write \(7 + x^4 - 3x^2 + x\) in standard form and state its degree. Show answerStandard form: \(x^4 - 3x^2 + x + 7\). Degree: 4 (quartic).
5. What is the degree of the polynomial \(4x^2 y^3 - xy + 6\)? (Hint: for multi-variable terms, add the exponents in each term.) Show answerThe term \(4x^2 y^3\) has total degree \(2 + 3 = 5\). The term \(-xy\) has total degree \(1 + 1 = 2\). The constant has degree 0. The polynomial's degree is 5.