Polynomial definitions, degree, standard form
High School Essentials • 9-12
Polynomials are the building blocks of algebra. From the simple expression 3x + 1 to the complex equation modeling a rocket's trajectory, polynomials appear throughout mathematics and science. This lesson establishes the vocabulary and foundational operations you need to work with them confidently.
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, where all exponents on the variables are whole numbers (0, 1, 2, 3, ...).
These ARE polynomials:
These are NOT polynomials:
| Name | Number of Terms | Examples |
|---|---|---|
| Monomial | 1 term | 7x2, -3, 4ab |
| Binomial | 2 terms | x + 5, 3y2 - 2y |
| Trinomial | 3 terms | x2 + 3x - 4 |
| Polynomial | 4+ terms (or general) | x3 - 2x2 + x - 1 |
The degree of a term is the sum of the exponents on its variables. The degree of a polynomial is the highest degree among all its terms.
Standard form means writing terms in descending order of degree, from highest to lowest.
Write 7 - 2x + 5x3 - x2 in standard form and state its degree.
This is a polynomial of degree 3 (also called a cubic) with 4 terms.
To add or subtract polynomials, combine like terms -- terms with exactly the same variable(s) raised to exactly the same power(s).
Add (3x2 + 5x - 4) + (2x2 - 3x + 7).
Subtract (4x3 - x + 6) - (2x3 + 3x2 - x + 1).
When subtracting polynomials, the subtraction sign applies to every term in the second polynomial. A frequent error is changing only the first term's sign. For (A) - (B - C), the result is A - B + C, not A - B - C.
When adding or subtracting polynomials, the degree of the result is at most the highest degree among the inputs -- but it could be lower if the leading terms cancel. For example, (3x2 + x) - (3x2 - 4) = x + 4, which has degree 1, not 2.
1. Classify 4x2 - 9 by number of terms and state its degree.
It has 2 terms, so it is a binomial. The highest exponent is 2, so the degree is 2.
2. Write 8 + 3x4 - x2 + 2x in standard form.
Arrange by descending degree: 3x4 - x2 + 2x + 8.
3. Add (6x2 - 3x + 1) + (-2x2 + 7x - 5).
(6x2 - 2x2) + (-3x + 7x) + (1 - 5) = 4x2 + 4x - 4.
4. Subtract (5x3 + 2x - 8) - (5x3 - x2 + 2x + 3).
Distribute: 5x3 + 2x - 8 - 5x3 + x2 - 2x - 3. Combine: x2 - 11.
5. Is 3x2 + 2/x a polynomial? Explain.
No. The term 2/x = 2x-1 has a negative exponent. Polynomials require all exponents to be non-negative whole numbers.
Polynomials are expressions with variables raised to whole-number exponents. They are classified by number of terms (monomial, binomial, trinomial) and by degree (the highest exponent). Standard form lists terms from highest to lowest degree. Adding and subtracting polynomials requires combining like terms, with special care to distribute negatives when subtracting.