Add, subtract, multiply polynomials, evaluate at x, or find roots. Enter coefficients for each term.
A polynomial is a mathematical expression made up of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples include 3x² + 2x − 5, x³ − 4x + 1, and 7 (a constant). Polynomials are fundamental in algebra, calculus, and virtually every area of mathematics and science.
Adding and subtracting polynomials means combining like terms — terms with the same variable and exponent. Multiplying polynomials uses the distributive property: every term in the first polynomial is multiplied by every term in the second, and the results are combined. Finding roots means finding the values of x where the polynomial equals zero — these are also called zeros or solutions.
The degree is the highest exponent of the variable in the polynomial. For example, 4x³ + 2x − 1 has degree 3. The degree determines many properties of the polynomial including the maximum number of roots it can have.
A polynomial of degree n has exactly n roots when counted with multiplicity and including complex roots. A quadratic (degree 2) has exactly 2 roots — they may be real, equal, or complex. A cubic has 3 roots, and so on.
When the discriminant of a quadratic (b² − 4ac) is negative, the roots involve the square root of a negative number, which produces complex numbers of the form a ± bi where i = √(−1). Complex roots always come in conjugate pairs.