Math
Multiplying Polynomials Calculator
Type two polynomials such as x^2 + 2x + 1 and x - 3 and the multiplying polynomials calculator distributes every term of the first over every term of the second.
The partial products are laid out as a grid, like terms are collected by degree, and the product is written in standard form. It handles any degrees, fraction coefficients, and monomials.
Use ^ or ² for powers. A single term such as 3x^2 is fine.
Same variable as the first polynomial.
Try an example
Result
Product
—
- Full statement
- —
- Distribution grid (rows: first factor terms)
- —
- All partial products
- —
- Number of partial products
- —
- Degree
- —
- Leading coefficient
- —
More details (3 more)
- Constant term
- —
- Terms in the product
- —
- Value at x = 1 (quick check)
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the multiplying polynomials calculator solves
To multiply polynomials, multiply every term of the first by every term of the second, then add the results and combine like terms. A polynomial with m terms times one with n terms gives m × n partial products before combining. The grid (box) method organizes those products in a table so none are missed.
Grid for (x^2 + 2x + 1)(x - 3)
| × | x | -3 |
|---|---|---|
| x^2 | x^3 | -3x^2 |
| 2x | 2x^2 | -6x |
| 1 | x | -3 |
Collect the cells by degree: x^3, then -3x^2 + 2x^2 = -x^2, then -6x + x = -5x, then -3. The product is x^3 - x^2 - 5x - 3.
How to use it
- Type the first polynomial and the second polynomial with ^ for powers.
- Read the product in standard form and the grid of partial products.
- Open the steps to see each multiplication and each group of like terms being combined.
How to read the answer
The product is written with descending powers and every like term combined. The number of partial products is the product of the term counts, but the final polynomial usually has fewer terms because like terms merge. The value at x = 1 equals the product of the sums of each factor's coefficients, a fast check.
Common mistakes and edge cases
- Multiplying The small raised number saying how many times to multiply the base by itself. instead of adding them: x^2 · x^3 = x^5, not x^6.
- Skipping a partial product. With 3 × 2 terms there must be 6 products before combining.
- Sign errors on negative terms: (-3)(-6x) = +18x.
- Combining terms with different degrees, such as x^2 and x.
Worked examples
Trinomial times binomial
(x^2 + 2x + 1)(x - 3) = x^3 - x^2 - 5x - 3
Product
x^3 - x^2 - 5x - 3
Monomial times polynomial
3x^2(2x^2 - x + 4) = 6x^4 - 3x^3 + 12x^2
Product
6x^4 - 3x^3 + 12x^2
Two trinomials
(x^2 + x + 1)(x^2 - x + 1) = x^4 + x^2 + 1
Product
x^4 + x^2 + 1
Fraction coefficients
((1/2)x + 3)(4x - 2) = 2x^2 + 11x - 6
Product
2x^2 + 11x - 6
Negative leading terms
(-2x + 1)(-x - 5) = 2x^2 + 9x - 5
Product
2x^2 + 9x - 5
Zero factor
Multiplying by 0 is rejected with a clear message
Product
Error
Frequently asked questions
How do you multiply two polynomials?+
Multiply each term of the first polynomial by each term of the second, add the exponents of matching variables, then combine like terms. The grid method lays the products out in a table so every pair is covered.
Does FOIL work for trinomials?+
FOIL is only the two-binomial case (4 products). For longer polynomials use full distribution or the grid: a trinomial times a binomial has 6 products, two trinomials have 9.
What is the degree of the product?+
The sum of the degrees of the factors, as long as neither factor is zero. A degree-2 times a degree-1 polynomial has degree 3.
How many terms will the product have?+
At most the product of the term counts, usually fewer because like terms combine. (x^2 + x + 1)(x^2 - x + 1) has 9 partial products but only 3 terms in the answer.
Can I multiply polynomials with fractions or decimals?+
Yes. Coefficients are converted to exact fractions, so (1/2)x × 4x = 2x^2 exactly.
How do I check a polynomial product?+
Substitute x = 1 (or another small number) into both factors and into your answer. The calculator lists the value at x = 1 for the product.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, Elementary Algebra 2e, Section 6.3 “Multiply Polynomials” — Distributive property, FOIL, and the vertical method.
Related calculators
Multiply Binomials Calculator
Multiply binomials with FOIL or expand any product of polynomials. Type (2x + 3)(x - 4) and see the First, Outer, Inner, Last products, like terms combined, and the expanded polynomial.
Box Method Calculator
Box method (area model) calculator that multiplies polynomials with an n × m box or factors a trinomial with the 2 × 2 box and ac method. Draws the box, labels every cell, and reads the factors off the edges.
Adding and Subtracting Polynomials Calculator
Add or subtract two polynomials by lining up like terms. Type each polynomial, see every column of like terms combined, the minus sign distributed, and the result in standard form with its degree and leading coefficient.
Polynomial Division Calculator
Divide polynomials by long division or synthetic division. Type both polynomials and get the quotient, remainder, the full synthetic-division tableau or long-division rows, the check, and every step.
Factoring Trinomials Calculator
Factor ax^2 + bx + c over the integers with the ac-method: the full factor-pair search table, splitting the middle term, grouping, and the roots when it does not factor.
Completing the Square Calculator
Complete the square on ax^2 + bx + c step by step: every algebraic line shown, exact fractions, vertex form a(x - h)^2 + k, the vertex, and the roots as simplified radicals.
Last updated: September 4, 2026