What Actually Qualifies as a Polynomial
A polynomial is a mathematical expression made up of one or more terms, where each term is a coefficient multiplied by a variable raised to a whole-number exponent. That last part is the key condition: the exponents have to be 0, 1, 2, 3, and so on. No fractions, no negatives, ever.
NOT polynomials:
3x⁻² + 1 (negative exponent)
x^(1/2) − 4 (fractional exponent = square root)
2/x + x (2/x = 2x⁻¹, negative exponent)
A standalone constant like 5 counts as a polynomial because 5 = 5x⁰, and 0 is a perfectly valid whole-number exponent. So constants are just the simplest possible case.
Terms, Coefficients, and What Degree Means
Each chunk of a polynomial separated by a + or − sign is called a term. Every term has two parts: a coefficient (the number out front) and a variable part (the variable with its exponent).
| Term | Coefficient | Variable Part | Degree of Term |
|---|---|---|---|
| 6x⁴ | 6 | x⁴ | 4 |
| −3x² | −3 | x² | 2 |
| x | 1 (implied) | x¹ | 1 |
| −9 | −9 | x⁰ (= 1) | 0 |
The degree of a polynomial is just the highest degree among all its terms. For 6x⁴ − 3x² + x − 9, the highest-degree term is 6x⁴, so the polynomial has degree 4. Each degree also has a name you'll see come up often:
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 11 |
| 1 | Linear | 2x + 5 |
| 2 | Quadratic | x² − 4x + 1 |
| 3 | Cubic | 3x³ + x² − 2 |
| 4 | Quartic | x⁴ − 7 |
| 5 | Quintic | 2x⁵ + x |
One Term, Two Terms, Three Terms
Polynomials also get names based on how many terms they contain:
- Monomial - exactly one term: 7x³ −4 5x²y
- Binomial - exactly two terms: x + 3 4x² − 9 a³ + b³
- Trinomial - exactly three terms: x² + 5x + 6 2a² − a + 1
For four or more terms, you just say polynomial. The term count doesn't change how you work with the expression, but the names come up constantly in factoring discussions, so it's worth knowing them.
−3x → monomial (1 term, degree 1 = linear)
x² − 16 → binomial (2 terms, degree 2 = quadratic)
4x² + 3x − 7 → trinomial (3 terms, degree 2 = quadratic)
Standard Form: Highest Degree First
A polynomial is in standard form when its terms are arranged in descending order of degree - highest-degree term first, constant last. It's not mathematically necessary, since addition is commutative and any order gives the same value. But it's like how you write a date: everyone agrees on a format so nobody gets confused. Standard form makes polynomials easy to compare, add, and factor without having to hunt around for matching terms.
Standard form: 2x³ + x² − 3x + 5
Not standard form: 7x − x⁴ + 2x² + 1
Standard form: −x⁴ + 2x² + 7x + 1
When you rearrange, carry the sign with each term. The term −x⁴ has a negative leading coefficient, which is completely fine. The leading coefficient is just whatever number is attached to the highest-degree term in standard form.
Also notice that −x⁴ + 2x² + 7x + 1 has no x³ term. That's perfectly fine - it just means the x³ coefficient is 0, so we don't write it. The degree is still 4 because the leading term is −x⁴.
How to Add and Subtract Polynomials
You can only combine terms that are exactly alike - same variable, same exponent. These are called like terms. The coefficients can differ; the variable part has to match exactly.
Unlike terms (cannot combine): 5x² and 5x 3x and 3x²y
To add or subtract two polynomials, drop the brackets (distributing a negative sign when subtracting), spot the like terms, and add or subtract their coefficients:
Group like terms:
x³ terms: 5x³ + x³ = 6x³
x² terms: −2x² + 3x² = x²
x terms: 4x − x = 3x
constants: −1 + 6 = 5
Result: 6x³ + x² + 3x + 5
Distribute the negative: 4x² + 3x − 2 − x² + 5x − 7
Combine like terms:
x²: 4x² − x² = 3x²
x: 3x + 5x = 8x
constant: −2 − 7 = −9
Result: 3x² + 8x − 9
The most common error in subtraction is forgetting to flip the sign on every term in the second polynomial, not just the first one. Writing out the distribution step explicitly, like above, is the easiest way to avoid that. Once you're solid on adding and subtracting, the next step is multiplication - that's where FOIL and the general distributive approach come in.