Multiplying Polynomials Calculator - Algebra Tool
Multiply any two polynomials by entering their coefficients as comma-separated lists. Get the fully expanded product polynomial instantly.
Multiplying Polynomials Calculator
Enter coefficients as comma-separated numbers starting from the constant term. For example, '1, 2, 3' represents 1 + 2x + 3x².
Enter comma-separated coefficients from constant term: e.g. '1, 2, 3' for 1 + 2x + 3x²
Enter comma-separated coefficients from constant term: e.g. '1, 2, 3' for 1 + 2x + 3x²
About the Multiplying Polynomials Calculator
A polynomial is an algebraic expression consisting of one or more terms, each composed of a coefficient multiplied by a non-negative integer power of a variable. For example, 3x² + 2x + 1 is a degree-2 polynomial (quadratic) with coefficients 3, 2, and 1. Multiplying two polynomials is a foundational operation in algebra, calculus, and many applied fields.
The standard method for multiplying polynomials is to distribute every term of the first polynomial over every term of the second polynomial and then collect like terms. If the first polynomial has m terms and the second has n terms, the multiplication produces m×n intermediate products. After combining terms with the same power of x, the result is a polynomial of degree equal to the sum of the degrees of the two inputs.
This calculator uses the coefficient-list representation: you enter the coefficients in ascending order of power, starting from the constant term. The input '1, 2, 3' represents 1 + 2x + 3x², with coefficients 1 (for x⁰), 2 (for x¹), and 3 (for x²). The multiplication is then performed using the discrete convolution algorithm, where each output coefficient is the sum of all products of input coefficients whose indices sum to the output index.
The FOIL method for binomials is a special case of polynomial multiplication where each polynomial has exactly two terms (m = n = 2), producing four intermediate products. For trinomials or higher-degree polynomials, the same distributive process applies, simply with more terms. For example, multiplying (1 + x + x²) by (1 + x) produces a degree-3 polynomial: 1 + 2x + 2x² + x³.
Practical applications of polynomial multiplication are broad. In signal processing, multiplying polynomials whose coefficients represent filter taps or signal samples is equivalent to convolution, the core operation in digital filtering. In computer algebra systems, polynomial multiplication is used to expand symbolic expressions in computer-aided design and symbolic differentiation. In combinatorics, generating functions are polynomials whose coefficients count combinatorial objects, and multiplying them combines the objects. Mastering this calculator and the underlying algorithm builds a foundation for further study in algebra, discrete mathematics, and numerical computation.
Polynomial Multiplication Examples
Each example shows the coefficient inputs, the reconstructed polynomials, and their product.
| Input Coefficients | Product | Notes |
|---|---|---|
| P₁: 1, 2 | P₂: 3, 4 | 3 + 10x + 8x² | (1+2x)(3+4x): distribute 1 and 2x |
| P₁: 1, 0, 1 | P₂: 1, 1 | 1 + x + x² + x³ | (1+x²)(1+x): degree 2 × degree 1 |
| P₁: 1, 1 | P₂: 1, −1 | 1 − x² | Difference of squares pattern |
| P₁: 1, 2, 1 | P₂: 1, 1 | 1 + 3x + 3x² + x³ | Trinomial times binomial |
How to Use the Calculator
- Enter the coefficients of the first polynomial in 'First Polynomial', separated by commas, starting with the constant term (e.g., '2, 3' for 2 + 3x).
- Enter the coefficients of the second polynomial in 'Second Polynomial' in the same format.
- Click 'Calculate Product' to see the expanded product polynomial.
- Read the result in standard polynomial notation — coefficients are listed from constant term to highest power.
- Click Reset to clear both fields and start a new multiplication.
Frequently Asked Questions
What order should I enter the coefficients?
Enter coefficients in ascending power order, starting from the constant term (power 0). For the polynomial 3 + 2x + x², enter '3, 2, 1'. For a polynomial starting with zero constant like x² + 4x, enter '0, 4, 1'.
What is the degree of the product polynomial?
The degree of the product is always the sum of the degrees of the two input polynomials. If you multiply a degree-2 polynomial by a degree-3 polynomial, the product has degree 5. The coefficient list of the product therefore has degree-of-P1 + degree-of-P2 + 1 entries.
How is polynomial multiplication related to FOIL?
FOIL is the special case of polynomial multiplication where both polynomials are binomials (degree 1 with two terms). FOIL produces four products; general polynomial multiplication produces m×n products and uses the same distributive principle extended to more terms.
Can I enter polynomials with negative coefficients?
Yes. Enter negative coefficients with a minus sign, such as '1, -3, 2' for 1 − 3x + 2x². Separate the numbers with commas or spaces; the calculator handles negative signs correctly.
What does the convolution algorithm do?
Polynomial multiplication is mathematically identical to discrete convolution. Each coefficient of the product at position k equals the sum of all pairs (c₁[i], c₂[j]) where i + j = k. This is the same operation used in digital signal processing to combine filter responses.