Multiplying Polynomials Calculator

Background.

Expanding (2x + 1)(4x + 3) by hand is a matter of FOIL: four products, collect like terms, done. Expanding a quartic times a cubic means twenty products and a real chance of dropping a sign or filing a term under the wrong power. The bookkeeping, not the algebra, is where the mistakes happen.

The rule behind every expansion is distribution. Each term of the first polynomial multiplies each term of the second, and a term in x^i times a term in x^j lands on x^(i+j). Gather everything that lands on the same power and you have the product. Written with coefficients alone, that is the convolution c_k = Σ a_i × b_j over all pairs with i + j = k, the same operation used to combine probability distributions, smooth digital signals and, minus the carrying, multiply whole numbers digit by digit.

Polynomials go in here as lists of coefficients with the constant first, so 1 + 2x is typed 1, 2 and 5 − x + 3x³ is typed 5, -1, 0, 3. The zero matters: it holds the place of the missing x² term. The product comes back in the same order, together with its value at an x you choose. That second number is a built-in check, because the product evaluated at any x must equal A(x) × B(x), which you can confirm on a pocket calculator.

Only single-variable polynomials with numeric coefficients are handled. Coefficients may be negative or decimal, but factoring, division and coefficients containing other letters, such as a or y, are outside the page's reach.

What is multiplying polynomials calculator?

Multiplying polynomials means expanding the product of two polynomials into a single polynomial in standard form. If A has degree m and B has degree n, the product has degree m + n, and each of its coefficients is the sum of every product of a coefficient from A with one from B whose powers add up to that exponent. The result is given as a coefficient list, constant term first, plus the product's numerical value at a chosen x.

How to use this calculator.

1. 01Type the coefficients of polynomial A from the constant term upward, separated by commas, spaces or semicolons. For 4x² + 3 enter 3, 0, 4, with a zero standing in for the missing x term.
2. 02Do the same for polynomial B. Each list needs at least two numbers, so a plain constant such as 5 goes in as 5, 0. Use an ordinary hyphen for negative signs.
3. 03Choose a value of x at which to evaluate the product, and use it as a check: the result should equal A(x) × B(x).
4. 04Read the product coefficients in the same constant-first order. A list of 3, 10, 8 means 8x² + 10x + 3.

The formula.

If A(x) = a₀ + a₁x + … + a_m x^m and B(x) = b₀ + b₁x + … + b_n x^n, the product is C(x) = c₀ + c₁x + … + c_(m+n) x^(m+n), where c_k = Σ a_i × b_j over every pair of indices with i + j = k.

In practice each coefficient of A is multiplied by each coefficient of B, (m + 1) × (n + 1) multiplications in all, and each product is added into the slot for the sum of the two powers. The lowest coefficient is always a₀b₀ and the highest a_m × b_n; the ones in between are sums. The page then evaluates C at your x by adding c_k × x^k over every k.

Two properties are worth knowing. The degree of the product is the sum of the degrees, so the coefficient list always has one entry fewer than the two input lists combined. And the product's roots are exactly the roots of A together with those of B, since C(x) is zero precisely when one of the factors is.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.