What is a Polynomial?
A polynomial is an expression built from variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. It sounds technical but you've been working with polynomials since you first learned algebra — every linear equation and quadratic is a polynomial.
In this lesson
1 What a Polynomial Is
A polynomial is a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. Examples: 3x² + 2x − 7, x⁴ − 5x, 6 (a constant is a polynomial of degree 0), and 2x³ − x² + 4x + 1.
Polynomials can only have non-negative integer exponents (0, 1, 2, 3...). Terms like x⁻¹, x^(1/2), or 1/x are not polynomial terms.
2 Anatomy: Terms, Coefficients, Degree
In the polynomial 4x³ − 2x² + 7x − 9:
3 Types by Degree and Number of Terms
By degree: degree 0 = constant, degree 1 = linear (2x + 3), degree 2 = quadratic (x² + 5), degree 3 = cubic (x³ − 2), degree 4 = quartic, degree 5 = quintic.
By number of terms: 1 term = monomial (3x²), 2 terms = binomial (x + 4), 3 terms = trinomial (x² + 2x + 1), 4+ terms = polynomial (general).
4 Adding, Subtracting, and Multiplying
Adding/Subtracting: combine like terms (same variable, same exponent).
Multiplying: use the distributive property — multiply each term in the first polynomial by each term in the second.
5 What Is NOT a Polynomial
These are NOT polynomials and why: 1/x = x⁻¹ (negative exponent), √x = x^(1/2) (fractional exponent), 2ˣ (variable in the exponent — this is exponential, not polynomial), x/y (division by a variable).
2ˣ is NOT a polynomial — it's an exponential function. The variable must be the base, not the exponent. x² is polynomial; 2ˣ is not.
Practice Problems
📚 Further Reading & Resources
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