7.1Introduction to Polynomials
Identify, classify, and write polynomials in standard form. Understand degree, leading coefficient, and the vocabulary that describes polynomial expressions.
Why This Matters
Polynomials are the building blocks of all algebraic expressions. You'll work with them in every advanced math course — from factoring in Algebra 2 to finding derivatives in Calculus. Understanding their structure now makes everything easier later.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What is a polynomial, and how do we classify and write polynomials in a standard way?
Lesson Overview
A polynomial is an algebraic expression made up of one or more terms, where each term is a product of a number (coefficient) and a variable raised to a whole-number exponent. Polynomials are classified by the number of terms (monomial, binomial, trinomial) and by their degree — the highest exponent in the expression. Writing a polynomial in standard form means ordering terms from the highest degree to the lowest.
Degree Classification
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 8 |
| 1 | Linear | 3x + 2 |
| 2 | Quadratic | x² − 5x + 6 |
| 3 | Cubic | 2x³ + x − 4 |
| 4 | Quartic | x⁴ − 3x² |
Monomial
7x²
1 term
Binomial
3x + 5
2 terms
Trinomial
x² + 4x − 6
3 terms
Anatomy of a Polynomial in Standard Form
−3x³ + x² + 5x + 7
−3x³
Degree 3 term
Leading coefficient: −3
x²
Degree 2 term
Coefficient: 1
5x
Degree 1 term
Coefficient: 5
7
Degree 0 (constant)
No variable
NOT Polynomials — Why?
Worked Examples
Write 5x − 3x³ + 7 + x² in standard form. Then state the degree and leading coefficient.
Identify the degree of each term: −3x³ (degree 3), x² (degree 2), 5x (degree 1), 7 (degree 0).
Order from highest to lowest degree: −3x³ + x² + 5x + 7.
Degree = 3 (highest exponent). Leading coefficient = −3 (coefficient of the first term).
Classify 4x² − 9 by number of terms and by degree.
Count the terms: 4x² and −9 → 2 terms.
Find the highest degree: x² has degree 2.
Two terms → binomial. Degree 2 → quadratic.
Is 3x⁻² + 5 a polynomial? Explain.
Check each term for whole-number exponents.
3x⁻² has a negative exponent (−2), which is not a whole number.
Polynomials require non-negative integer exponents only.
Write the polynomial with terms: degree-2 term with coefficient 6, degree-0 term with coefficient −4, and degree-1 term with coefficient 1. State the number of terms and classify by degree.
Write each term: 6x², x (coefficient 1 is usually omitted), −4.
Order in standard form (highest to lowest degree): 6x² + x − 4.
Three terms → trinomial. Highest degree is 2 → quadratic.
Identify the degree, leading coefficient, and number of terms of: −7x⁴ + 2x³ − x + 11
The polynomial is already in standard form (degrees: 4, 3, 1, 0).
Highest degree = 4. Leading coefficient = −7 (coefficient of x⁴).
Count terms: −7x⁴, 2x³, −x, 11 → 4 terms.
Guided Practice
Guided Practice Video: Introduction to Polynomials
Watch the guided practice walkthrough for introduction to polynomials, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Write 9 − 4x + 2x³ in standard form. State the degree and leading coefficient.
Hint: Find the degree of each term, then order from highest to lowest.
Classify x² + 3x − 10 by number of terms and by degree.
Hint: Count the terms first, then find the highest exponent.
Is 5/x + 3 a polynomial? Explain why or why not.
Hint: Rewrite 5/x as 5x⁻¹. What is the exponent? Is it a whole number?
A polynomial has 3 terms, degree 4, and leading coefficient −2. Write one possible polynomial in standard form.
Hint: The first term must be −2x⁴. You choose the other two terms (any degree lower than 4).
Identify the degree, leading coefficient, and classify by number of terms: 8x⁵ − 3x² + x
Hint: Find the highest exponent for degree, the coefficient of that term for leading coefficient, and count terms.
Key Vocabulary
Polynomial
An expression with one or more terms, each consisting of a coefficient and a variable raised to a whole-number (non-negative integer) exponent.
Term
A single number, variable, or product of numbers and variables. Terms are separated by addition or subtraction.
Coefficient
The numerical factor of a term. In 5x², the coefficient is 5.
Degree of a Polynomial
The highest degree among all terms in the polynomial.
Leading Coefficient
The coefficient of the term with the highest degree when the polynomial is in standard form.
Standard Form
A polynomial written with terms in descending order of degree (highest to lowest).
Monomial / Binomial / Trinomial
Polynomials with exactly 1 / 2 / 3 terms respectively.
Constant
A term with no variable (degree 0). Example: 8. A nonzero constant is a degree-0 monomial.
Interactive Practice — 5 Questions
Which expression is NOT a polynomial?
What is the degree of 6x⁴ − 3x² + x − 8?
Which polynomial is written in standard form?
What is the leading coefficient of −4x³ + 7x − 1?
How is x² + 5x − 6 classified by number of terms?
Independent Practice
Independent Practice
Write each in standard form, then state the degree and leading coefficient: (a) 3 − x² + 4x (b) 7x + x³ − 2x² + 5. Show all steps.
Classify each by number of terms AND by degree: (a) 6x (b) x² − 4 (c) 2x³ + x − 8 (d) 9.
Determine whether each is a polynomial. If not, explain why: (a) x^(1/2) + 3 (b) 4x² − 3x + 1 (c) 2/x + 5.
Write: (a) a monomial of degree 5 with leading coefficient −3 (b) a trinomial of degree 3 in standard form (c) a binomial of degree 4.
Identify the degree and leading coefficient of −9x⁶ + 4x³ − x + 2. Then classify by number of terms. A student writes 2x + 5x³ − x² and says it is in standard form — is the student correct? Rewrite it correctly.
Common Mistakes
Counting the number of terms to find the degree — e.g., saying a trinomial has degree 3.
Degree = the highest exponent on the variable. A trinomial has 3 terms, but its degree depends on its exponents.
Writing standard form in ascending order (lowest to highest degree).
Standard form goes from highest to lowest degree: 3x⁴ − 2x² + x − 5.
Forgetting that the leading coefficient is the coefficient of the highest-degree term, not the first term as written.
Rewrite in standard form first, then identify the leading coefficient as the coefficient of the highest-degree term.
Forgetting the sign of the leading coefficient — e.g., saying the leading coefficient of −3x³ + x is 3.
The sign is part of the coefficient. In −3x³ + x, the leading coefficient is −3, not 3.
Math Tips
Standard form: always write terms from highest degree to lowest — e.g., 3x² + 5x − 2, not 5x − 2 + 3x².
Degree of a polynomial = the largest exponent. For 4x³ − 2x + 7, the degree is 3.
Leading coefficient = the coefficient of the first term in standard form (include the sign!).
A constant like 9 is a degree-0 monomial — it is still a valid polynomial.
NOT polynomials: negative exponents (x⁻²), fractional exponents (x^(1/2)), or variables in the denominator (1/x).