Polynomial Division Calculator

Polynomial division separates a dividend polynomial into a quotient multiplied by a divisor plus a remainder. It extends familiar whole-number long division to expressions containing powers of x. If P(x) is divided by D(x), the result satisfies P(x) = D(x)Q(x) + R(x), where Q(x) is the quotient and R(x) is the remainder. The degree of a nonzero remainder must be lower than the degree of the divisor. The long-division process begins by arranging both polynomials in descending powers. Missing powers are represented by zero coefficients, which is essential for keeping columns aligned. Divide the leading term of the current dividend by the leading term of the divisor. That value becomes the next quotient term. Multiply the entire divisor by it, subtract the product, and repeat with the new leading term until the remaining polynomial has lower degree than the divisor. This calculator performs those same operations on coefficient arrays. It accepts integer or decimal coefficients, x terms, nonnegative whole-number exponents, and omitted coefficients of one. For example, x^3 means 1x^3, and -x means -1x. Terms may be entered in any order because the parser combines like powers and inserts missing coefficients before division. An asterisk between a coefficient and x is optional. An exact division has remainder zero. For example, x^3 - 1 divided by x - 1 gives x^2 + x + 1 because multiplying that quotient by x - 1 recreates the dividend. A nonzero remainder is not an error; it means the divisor is not a polynomial factor of the dividend. The result can also be written as the quotient plus the remainder divided by the divisor. When the divisor is linear, polynomial long division produces the same result as synthetic division, although synthetic division is a shorter coefficient-only procedure. The Remainder Theorem provides another useful check: dividing P(x) by x - c produces remainder P(c). If that remainder is zero, c is a root and x - c is a factor. Polynomial division is used to simplify rational expressions, factor polynomials, find roots, identify asymptotes, integrate rational functions, and verify algebra exercises. After calculating, multiply the displayed quotient by the original divisor and add the remainder. The identity shown with the result summarizes that check and makes it easier to catch a copied sign or coefficient.