Polynomial Multiplication Calculator

Enter an expression with the polynomials to multiply. You can use multiple factors and grouping symbols (parentheses, brackets, braces).

Quick Examples

Rate this tool

How to Use the Calculator

This online polynomial multiplication calculator is an algebraic tool designed to solve the product of expressions of any degree. The mathematical solver not only provides the simplified resulting polynomial but also generates the step-by-step expansion explaining the application of the distributive property.

To perform the calculation, follow these instructions:

  1. Enter the complete expression: Type the polynomials you want to multiply, structured by parentheses, into the single input field, for example: (x+2)(x-3). The system supports monomials, binomials, trinomials, and polynomials with any number of terms or exponents. You can enter coefficients that are integers, rational numbers (fractions), or irrational values expressed as roots of any index (e.g., square or cube roots). The tool will maintain the exact symbolic format to prevent decimal rounding errors.
  2. Multiplication of multiple factors: Do not limit yourself to just two polynomials. The solver is equipped to process the consecutive multiplication of several factors, allowing inputs like (x+2)(x+3)(x2+5) directly.
  3. Step-by-step solution: Once the calculation is initiated, the system displays the procedure applying the distributive property. If the expression contains more than two factors, the algorithm solves the problem associatively, two at a time: it first multiplies the first two polynomials, simplifies the intermediate result, and then multiplies it by the third polynomial, repeating this process sequentially until all factors are covered.
  4. Simplified and ordered result: Upon concluding the procedure, the tool combines like terms and displays the final polynomial completely reduced and ordered from highest to lowest exponent.

Solved Exercises

The following are examples of problems solved by the calculator.

Multiply the binomials x + 3 and x - 2.

Result

$$ x^{2} + x - 6 $$

Step-by-step solution

1. Analyze the expression.

The operation to solve is:

$$ (x + 3)(x - 2) $$

2. Apply the distributive property.

Multiply each term of the first polynomial by each term of the second:

$$ (x + 3)(x - 2) \\[1.5em] x(x) + x(-2) + 3(x) + 3(-2) \\[1.5em] x^{2} - 2 x + 3 x - 6 $$

Combine like terms:

$$ x^{2} + x - 6 $$
Find the product of the monomial x2 and the trinomial x2 + 2x + 1.

Result

$$ x^{4} + 2 x^{3} + x^{2} $$

Step-by-step solution

1. Analyze the expression.

The operation to solve is:

$$ x^{2}(x^{2} + 2 x + 1) $$

2. Apply the distributive property.

Multiply each term of the first polynomial by each term of the second:

$$ x^{2}(x^{2} + 2 x + 1) \\[1.5em] x^{2}(x^{2}) + x^{2}(2 x) + x^{2}(1) \\[1.5em] x^{4} + 2 x^{3} + x^{2} $$
Calculate the product of the two polynomials 2x2 - 5 and 3x3 + 4.

Result

$$ 6 x^{5} - 15 x^{4} + 8 x^{2} - 20 x $$

Step-by-step solution

1. Analyze the expression.

The operation to solve is:

$$ (2 x^{2} - 5 x)(3 x^{3} + 4) $$

2. Apply the distributive property.

Multiply each term of the first polynomial by each term of the second:

$$ (2 x^{2} - 5 x)(3 x^{3} + 4) \\[1.5em] 2 x^{2}(3 x^{3}) + 2 x^{2}(4) - 5 x(3 x^{3}) - 5 x(4) \\[1.5em] 6 x^{5} + 8 x^{2} - 15 x^{4} - 20 x $$

Combine like terms:

$$ 6 x^{5} - 15 x^{4} + 8 x^{2} - 20 x $$
Find the result of the multiplication \((x-\sqrt{2})\cdot (x+\sqrt{2}).\)

Result

$$ x^{2} - 2 $$

Step-by-step solution

1. Analyze the expression.

The operation to solve is:

$$ \left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right) $$

2. Apply the distributive property.

Multiply each term of the first polynomial by each term of the second:

$$ \left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right) \\[1.5em] x(x) + x\left(\sqrt{2}\right) - \sqrt{2}(x) - \sqrt{2}\left(\sqrt{2}\right) \\[1.5em] x^{2} + \sqrt{2} x - \sqrt{2} x - 2 $$

Combine like terms:

$$ x^{2} - 2 $$
Determine the result of multiplying the three polynomials (x2 - x), (x + 2), and (x - 3).

Result

$$ x^{4} - 2 x^{3} - 5 x^{2} + 6 x $$

Step-by-step solution

1. Analyze the expression.

The operation to solve is:

$$ (x^{2} - x)(x + 2)(x - 3) $$

2. Apply the distributive property.

Multiply each term of the first polynomial by each term of the second:

$$ (x^{2} - x)(x + 2) \\[1.5em] x^{2}(x) + x^{2}(2) - x(x) - x(2) \\[1.5em] x^{3} + 2 x^{2} - x^{2} - 2 x $$

Combine like terms:

$$ x^{3} + x^{2} - 2 x $$

Multiply the obtained result by the next polynomial in the expression:

$$ (x^{3} + x^{2} - 2 x)(x - 3) \\[1.5em] x^{3}(x) + x^{3}(-3) + x^{2}(x) + x^{2}(-3) - 2 x(x) - 2 x(-3) \\[1.5em] x^{4} - 3 x^{3} + x^{3} - 3 x^{2} - 2 x^{2} + 6 x $$

Combine like terms:

$$ x^{4} - 2 x^{3} - 5 x^{2} + 6 x $$

Related Tools

Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (5 days ago)