What is a Polynomial?
A polynomial is an algebraic expression made up of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. Polynomials range from simple linear expressions to high-degree curves, and they model everything from projectile paths to business cost functions.
A polynomial is an expression of the form aₙxⁿ + ... + a₁x + a₀, where each term has a variable raised to a whole-number power multiplied by a coefficient, and the highest exponent is called the degree.
Try it: interactive calculator
Step-by-step worked examples
Add the polynomials (3x² + 2x − 1) and (x² − 5x + 4).
Group like terms: (3x²+x²) + (2x−5x) + (−1+4) Combine: 4x² − 3x + 3 Result: 4x² − 3x + 3
Multiply (x + 2)(x − 3).
Apply FOIL: x·x + x·(−3) + 2·x + 2·(−3) Simplify: x² − 3x + 2x − 6 Combine: x² − x − 6
Find the degree of 5x⁴ − 3x² + 7x − 1.
Identify the exponents: 4, 2, 1, 0 The degree is the highest exponent Degree = 4
Flashcards
Quick quiz
Q1.What is the degree of 7x³ − 2x + 5?
Q2.Simplify (2x + 3) + (4x − 1).
Q3.Which of these is NOT a polynomial term?
Q4.What is the leading coefficient of −4x⁵ + 2x² − 9?
The full card deck, worked steps and AI-tutor support for “What is a Polynomial?” are in Notek — study by hand before your exam.
Common mistakes
Thinking any expression with x is a polynomial. — Correct: Expressions with negative exponents, fractional exponents, or x in a denominator are NOT polynomials.
Adding unlike terms together, e.g. 3x² + 2x = 5x². — Correct: Only like terms (same variable and exponent) can be combined; 3x² and 2x stay separate.
Confusing degree with the number of terms. — Correct: Degree is the highest exponent, not how many terms the polynomial has.
Forgetting to distribute the negative sign when subtracting polynomials. — Correct: Subtracting a polynomial means flipping the sign of every term inside the parentheses.
FAQ
What is a polynomial?
A polynomial is an expression built from variables with whole-number exponents and coefficients, combined by addition, subtraction, and multiplication.
What is the formula for evaluating a polynomial?
Substitute the given x-value into each term aₙxⁿ and add the results together.
What are examples of polynomials?
x² − 4, 3x³ + 2x − 1, and 7 are all polynomials (of degree 2, 3, and 0 respectively).
How do you calculate the degree of a polynomial?
Find the term with the highest exponent on its variable — that exponent is the degree.