Multiplying Polynomials Calculator

Multiplying polynomials relies on the distributive property of multiplication over addition, often visualized through the FOIL method for binomials or extended distribution for higher-degree polynomials. If you have polynomial P(x) = a_n x^n + ... + a_0 and polynomial Q(x) = b_m x^m + ... + b_0, their product R(x) is found by multiplying every term in P(x) by every term in Q(x) and then combining like terms. The degree of the resulting product polynomial is always equal to the sum of the degrees of the two input polynomials (Deg(P) + Deg(Q)).

Let us find the product of two simple polynomials: P(x) = 2x + 3 and Q(x) = x^2 + 4x + 1. Here, the coefficients for P(x) are a_1 = 2 and a_0 = 3. The coefficients for Q(x) are b_2 = 1, b_1 = 4, and b_0 = 1. First, multiply each term of Q(x) by 2x: (2x)(x^2) + (2x)(4x) + (2x)(1) = 2x^3 + 8x^2 + 2x. Next, multiply each term of Q(x) by 3: (3)(x^2) + (3)(4x) + (3)(1) = 3x^2 + 12x + 3. Finally, combine all the terms by grouping powers of x together: 2x^3 + (8x^2 + 3x^2) + (2x + 12x) + 3, which simplifies to the final product R(x) = 2x^3 + 11x^2 + 14x + 3.

Always arrange your input polynomials in standard descending order of their exponents before setting up your calculation to prevent coefficient placement errors. Pay close attention to negative signs during the distribution phase, as sign drops are the most frequent source of calculation mistakes in algebra. Use this calculator to verify your manual homework steps, ensuring you understand the underlying distribution rather than just relying on the final output.