A polynomial is a mathematical expression built from variables, coefficients, and non-negative integer exponents, combined via addition and subtraction. General form:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₁x + a₀
Where aₙ…a₀ are coefficients and n is the degree — the highest non-zero exponent.
Interactive Polynomial Builder
−5 to 5
P(x) = x³ − 3x² + 2
Graph — click to inspect a point
Properties
Degree
3
Leading Coeff.
1
Type
Cubic
y-intercept
2
Standard Form
x³ − 3x² + 2
What is the degree of 4x² − 2x + 7?
Key Terminology
Term — a single product of coefficient × power (e.g. 3x²)
Coefficient — the numeric factor of a term
Degree — the highest exponent present
Leading coefficient — coefficient of the highest-degree term
Constant term — the term with no variable
Operations Module 02
Adding & Subtracting Polynomials
Combine like terms — those sharing the same variable and exponent. Only matching powers interact.
(3x² + 2x − 5) + (x² − 4x + 7) = 4x² − 2x + 2
Multiplying Polynomials
Apply the distributive property — every term of the first polynomial multiplies every term of the second, then collect like terms.
Choose Operation
P₁ — First Polynomial
P₂ — Second Polynomial
(2x² + 3x + 1) + (x² − 2x + 4) = 3x² + x + 5
Visualisation — P₁ (cyan) P₂ (magenta) Result (yellow)
Step-by-Step Breakdown
Graphing Module 03
End Behaviour
As x → ±∞, the highest-degree term dominates. The sign and parity of the degree determine tail direction:
Even degree, positive leading coeff. — both ends rise (∪ shape)
Even degree, negative leading coeff. — both ends fall (∩ shape)
Odd degree, positive leading coeff. — falls left, rises right
Odd degree, negative leading coeff. — rises left, falls right
Explore Graph Features
Interactive Graph
Graph Analysis
Roots
—
y-intercept
2
End Behaviour
—
Max Turning Pts
2
End behaviour of a cubic with a positive leading coefficient?
Theorems Module 04
Remainder Theorem
When P(x) is divided by (x − c), the remainder equals P(c).