Multiplying Polynomials Calculator

Multiply two polynomials and get the resulting polynomial coefficients instantly.

Enter the coefficients of two polynomials to compute their product. This tool uses the distributive property (convolution of coefficients) to find the solution.

Enter comma-separated or space-separated numbers.

Enter comma-separated or space-separated numbers.

Examples

Click on an example to load it into the calculator.

Multiply Two Binomials (FOIL)

multiplication

Multiplying (x + 2) by (x + 3). Coefficients for x+2 are [2, 1]. Coefficients for x+3 are [3, 1].

P₁: [2, 1]

P₂: [3, 1]

Binomial and Trinomial

multiplication

Multiplying (2x - 3) by (x² + 4x - 5). Coeffs: [-3, 2] and [-5, 4, 1].

P₁: [-3, 2]

P₂: [-5, 4, 1]

Multiplying by a Constant

multiplication

Multiplying (3x² - x + 1) by 4. Coeffs: [1, -1, 3] and [4].

P₁: [1, -1, 3]

P₂: [4]

Two Trinomials

multiplication

Multiplying (x² + 2x + 1) by (x² - 3x + 2). Coeffs: [1, 2, 1] and [2, -3, 1].

P₁: [1, 2, 1]

P₂: [2, -3, 1]

Other Titles
Understanding Multiplying Polynomials: A Comprehensive Guide
Master the art of multiplying polynomials, from simple binomials to complex expressions, and understand its core principles and applications.

What is Polynomial Multiplication? Core Concepts

  • The process of applying the distributive property multiple times
  • Combining terms to form a new, higher-degree polynomial
  • Foundation for solving algebraic equations and modeling systems
Polynomial multiplication is a fundamental operation in algebra that involves finding the product of two or more polynomials. The core principle is to use the distributive property repeatedly, ensuring that every term in the first polynomial is multiplied by every term in the second polynomial.
When you multiply two polynomials, the result is a new polynomial whose degree is the sum of the degrees of the original polynomials. The coefficients of this new polynomial are found by combining the products of the coefficients of the original terms. This process is mathematically equivalent to the convolution of the coefficient sequences.
The Distributive Property
For example, to multiply (ax + b) by (cx + d), you distribute each term: ax(cx + d) + b*(cx + d) = acx² + adx + bcx + bd. Combining like terms gives acx² + (ad + bc)x + bd.

Basic Multiplication Examples

  • (x + 1) * (x + 2) = x² + 3x + 2
  • Coefficients [1, 1] * [2, 1] results in coefficients [2, 3, 1]
  • Multiplying a polynomial by a constant (a monomial of degree 0) scales all its coefficients.

Step-by-Step Guide to Using the Multiplying Polynomials Calculator

  • Learn the correct format for entering polynomial coefficients
  • Execute the calculation and understand the output
  • Use the reset and example features for efficient workflow
Our calculator simplifies polynomial multiplication into a few easy steps, providing accurate results for your algebra homework, engineering calculations, or scientific research.
Input Guidelines:
  • Coefficient Format: Enter the coefficients for each polynomial separated by commas (e.g., '3, 0, -1' for 3x² - 1) or spaces ('3 0 -1').
  • Order of Coefficients: The coefficients must be entered from the lowest degree to the highest. For a polynomial like 2x³ + 4x - 5, you would enter '-5, 4, 0, 2' (note the '0' for the missing x² term).
Calculation and Results:
  • Calculate: Click the 'Calculate Product' button to perform the multiplication.
  • Resulting Coefficients: The output shows the coefficients of the resulting polynomial, also ordered from lowest to highest degree.
  • Formatted Polynomial: A human-readable version of the resulting polynomial is displayed for clarity.

Practical Usage Examples

  • Input P₁: '1, 1', P₂: '1, 1' → Result: 1 + 2x + x²
  • Input P₁: '2, -1', P₂: '3, 2, 1' → Result: 6 + x - x²

Real-World Applications of Multiplying Polynomials

  • Engineering: Modeling signals and designing systems
  • Computer Graphics: Creating curves and surfaces
  • Cryptography: Building secure encryption algorithms
  • Financial Modeling: Predicting growth and analyzing trends
Polynomial multiplication is not just an abstract algebraic concept; it's a powerful tool used in various fields of science, technology, and finance.
Signal Processing and System Design:
In engineering, the characteristics of linear time-invariant (LTI) systems are described by polynomials. Multiplying these polynomials is equivalent to cascading systems, allowing engineers to predict the overall output.
Computer Graphics and Geometry:
Polynomials define curves and surfaces, such as Bézier curves used in vector graphics and font design. Multiplying them can help in geometric modeling and creating complex shapes.
Cryptography:
Advanced encryption standards, particularly those based on Galois Fields (finite fields), rely heavily on polynomial arithmetic, including multiplication, to ensure data security.

Industry Applications

  • Modeling area: (length+a)*(width+b) is a polynomial multiplication problem.
  • Bézier curve calculations involve products of Bernstein basis polynomials.
  • Error-correcting codes use polynomial multiplication over finite fields to encode and decode data.

Common Methods: FOIL, Grid, and Vertical Multiplication

  • The FOIL method for multiplying two binomials
  • The Grid (or Box) method for organizing terms
  • The Vertical method, similar to multi-digit number multiplication
While our calculator provides an instant answer, understanding the manual methods is crucial for building a strong foundation in algebra. Each method is a systematic way of applying the distributive property.
The FOIL Method:
FOIL is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. For (ax+b)(cx+d), you compute (ax)(cx) + (ax)(d) + (b)(cx) + (b)(d). It's a special case of the general distributive method.
The Grid (Box) Method:
This method uses a grid to organize the products of terms. Write the terms of one polynomial along the top and the terms of the other down the side. Fill in each cell with the product of the corresponding row and column terms. Finally, combine the like terms (often found on the diagonals).
The Vertical Method:
This looks very similar to long multiplication with numbers. You write one polynomial above the other and multiply the top polynomial by each term of the bottom polynomial, lining up like terms vertically before adding them up.

Manual Calculation Techniques

  • FOIL: (x+2)(x+3) = (x*x) + (x*3) + (2*x) + (2*3) = x² + 5x + 6.
  • Vertical Method: (x²+2x+1) multiplied by (x-1) requires two rows of partial products before summing.
  • The Grid Method is excellent for visual learners and helps avoid missing terms.

Mathematical Derivation: Multiplication as Coefficient Convolution

  • Representing polynomials as sequences of coefficients
  • Understanding the formal definition of discrete convolution
  • Connecting the convolution formula to the distributive property
The computational engine behind this calculator is an elegant mathematical concept: discrete convolution. Understanding this connection reveals the deep structure of polynomial arithmetic.
Polynomials as Vectors:
A polynomial P(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ can be uniquely represented by its vector (or sequence) of coefficients [a₀, a₁, a₂, ..., aₙ].
Convolution Formula:
Let P₁(x) have coefficients A = [a₀, a₁, ...] and P₂(x) have coefficients B = [b₀, b₁, ...]. The product P(x) = P₁(x)P₂(x) has coefficients C = [c₀, c₁, ...], where each cₖ is given by the discrete convolution formula: cₖ = Σᵢ aᵢ * bₖ₋ᵢ, where the sum is over all valid indices i.
For example, the coefficient of the x² term (c₂) is the sum of all products of coefficients where the degrees add up to 2: a₀b₂ + a₁b₁ + a₂b₀. This is precisely what happens when you gather like terms after distributing.

Derivation Through Convolution

  • P₁ = x+2 → [2, 1], P₂ = x+3 → [3, 1].
  • c₀ = a₀b₀ = 2 * 3 = 6.
  • c₁ = a₀b₁ + a₁b₀ = (2 * 1) + (1 * 3) = 5.
  • c₂ = a₁b₁ = 1 * 1 = 1.
  • Resulting coefficients C = [6, 5, 1], which corresponds to x² + 5x + 6.