Enter two polynomials using their coefficients (from constant term to highest degree). Instantly get the product in standard form, the complete coefficient array, and a detailed breakdown of the multiplication using the distributive property.
Multiplying polynomials is a fundamental algebraic operation that combines two or more polynomials into a single polynomial. It relies on the distributive property (also known as the FOIL method for binomials). The product is obtained by multiplying each term of the first polynomial by every term of the second, then combining like terms (same exponent). Polynomial multiplication appears everywhere: from simple area calculations to advanced physics and engineering models.
(a₀ + a₁x + a₂x² + …)(b₀ + b₁x + b₂x² + …) = Σₖ ( Σᵢⱼ with i+j=k aᵢ·bⱼ ) xᵏ
(convolution of coefficient sequences)
The rules for multiplying polynomials were known in ancient Babylonian and Greek mathematics, but the symbolic notation we use today was developed in the 16th and 17th centuries by mathematicians like François Viète and René Descartes. The distributive property was formally stated by Peacock and others in the 19th century. Modern algebra views polynomial multiplication as a convolution – an operation that also appears in signal processing, probability, and number theory.
Given two polynomials represented by coefficient arrays A = [a₀, a₁, …, am] and B = [b₀, b₁, …, bn] (where index = degree), their product C has degree m+n and coefficients:
ck = Σ ai·bj for all i+j = k, 0 ≤ i ≤ m, 0 ≤ j ≤ n.
This is exactly the discrete convolution operation. For example, multiplying (1 + 2x + x²) by (1 + x) yields coefficients: c₀ = 1·1 = 1, c₁ = 1·1 + 2·1 = 3, c₂ = 2·1 + 1·1 = 3, c₃ = 1·1 = 1, resulting in 1 + 3x + 3x² + x³. Our calculator automates this convolution while displaying each intermediate product.
All examples below are verified with this calculator – click the buttons to test.
| Polynomial A (coeffs) | Polynomial B (coeffs) | Product (standard form) | Degree |
|---|---|---|---|
| [1,1] (x+1) | [1,2] (x+2) | x² + 3x + 2 | 2 |
| [1,-3] (x-3) | [1,3] (x+3) | x² - 9 | 2 |
| [2,1] (2x+1) | [1,3,-2] (x²+3x-2) | 2x³ + 7x² - x - 2 | 3 |
| [5] (constant 5) | [2,0,-1] (2x²-1) | 10x² - 5 | 2 |
| [1,0,-4] (x²-4) | [1,0,2] (x²+2) | x⁴ - 2x² - 8 | 4 |
A company models its revenue R as a product of price p and quantity sold q. If price is given by p(x) = 100 - 2x (linear demand) and quantity q(x) = 50 + 3x (seasonal factor), then revenue R(x) = p(x)·q(x) = (100 - 2x)(50 + 3x) = 5000 + 300x - 100x - 6x² = 5000 + 200x - 6x². Our calculator quickly expands such expressions, helping analysts identify turning points and optimize pricing.
Polynomial multiplication is closely related to:
function multiplyPoly(coeffA, coeffB) {
let result = new Array(coeffA.length + coeffB.length - 1).fill(0);
for (let i = 0; i < coeffA.length; i++) {
for (let j = 0; j < coeffB.length; j++) {
result[i + j] += coeffA[i] * coeffB[j];
}
}
// Trim trailing zeros (optional)
while (result.length > 1 && Math.abs(result[result.length-1]) < 1e-12) result.pop();
return result;
}
1,0,-4 (constant -4, then 0x, then 1x²).