Polynomial Division Calculator

Polynomial division operates similarly to traditional long division with integers, but involves algebraic terms sorted by descending powers of the variable. Given a dividend polynomial P(x) and a divisor polynomial Q(x), the algorithm seeks a quotient Q_out(x) and a remainder R(x) such that P(x) = Q(x) × Q_out(x) + R(x), where the degree of the remainder is strictly less than the degree of the divisor. The calculator processes coefficients from highest to lowest degree (A0 through A6 for the dividend, and B0 through B6 for the divisor) to execute this systematic reduction step by step.

Let us divide the dividend polynomial P(x) = 2x^3 + 3x^2 - x - 5 by the divisor polynomial Q(x) = x^2 + x - 2. First, we divide the leading term of the dividend (2x^3) by the leading term of the divisor (x^2), yielding the first term of our quotient: 2x. Next, we multiply this quotient term by the entire divisor: 2x(x^2 + x - 2) = 2x^3 + 2x^4 - 4x. Subtracting this product from P(x) gives a new remainder polynomial: x^2 + 3x - 5. Repeating the process, we divide x^2 by x^2 to get +1. Multiplying 1 by Q(x) and subtracting yields the final remainder of 2x - 3. Thus, our final quotient is 2x + 1 with a remainder of 2x - 3.

To ensure accurate results every time, always verify that your input polynomials are written in standard form with descending exponents. If any intermediate degree term is missing from your expression (for example, if you have x^3 - 4 instead of x^3 + 0x^2 + 0x - 4), make sure to account for that missing coefficient as zero. Double-checking your degree specifications (DegP and DegQ) will prevent misalignments and ensure the division algorithm executes cleanly.