Multiplying Polynomials Calculator
Enter two polynomials as comma-separated coefficients (highest degree first) and get their fully expanded product with each distribution step shown.
Distribute x^2 across (x - 1)
Distribute 2x across (x - 1)
Distribute 1 across (x - 1)
Combine all like terms
How does this calculator work?
Polynomial multiplication distributes every term of P over every term of Q, then combines like terms. Enter coefficients highest-to-lowest as comma-separated values (e.g. "1,2,1" = x²+2x+1). The product has degree equal to the sum of the input degrees. Each distribution step is shown for verification.
Formula
How this is calculated
Polynomial multiplication applies the distributive property: every term of the first polynomial is multiplied by every term of the second, and the partial products sharing the same degree are then combined as like terms. For polynomials of degree m and n the product always has degree m+n (assuming leading coefficients are non-zero).
Coefficients are entered from the highest to the lowest degree, separated by commas. For example, "1,2,1" represents x² + 2x + 1 and "1,-1" represents x − 1. The calculator distributes each coefficient of the first polynomial across the entire second polynomial, collects the results degree by degree, and displays the combined product polynomial. Each distribution step is shown so you can follow the working.
All arithmetic is exact floating-point. Very large polynomials (more than 15 coefficients each) are rejected to prevent slow computation. Coefficients can be any real numbers; integer coefficients give exact integer results. The degree of the product equals the sum of the degrees of the two input polynomials.
Frequently asked questions
Enter 0 for any missing coefficient. For example, x³ + 1 (missing the x² and x terms) should be entered as "1,0,0,1". The calculator handles zero coefficients correctly and omits the corresponding terms from the output.
If polynomial P has degree m and polynomial Q has degree n, their product P·Q has degree m+n (provided the leading coefficients are both non-zero). For example, multiplying a degree-2 polynomial by a degree-3 polynomial gives a degree-5 result with up to 6 terms.
Yes — any real number coefficient is accepted. Enter decimals directly (e.g. "0.5,1.25,-3"). Fractional inputs should be converted to decimals first. Results are shown to up to four decimal places; very small rounding errors can appear when the true coefficients are not exactly representable in floating-point.
Also known as
TG we-Calculate Editorial Team. (2026). Multiplying Polynomials Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/multiplying-polynomials-calculator
TG we-Calculate Editorial Team. "Multiplying Polynomials Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/multiplying-polynomials-calculator.
TG we-Calculate Editorial Team, "Multiplying Polynomials Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/multiplying-polynomials-calculator
@misc{wecalculate_multiplying_polynomials_calculator, title = {Multiplying Polynomials Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/multiplying-polynomials-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?