Multiplying Polynomials Calculator

Enter the coefficients of two polynomials to multiply them with the distributive property and get the fully expanded, like-terms-combined product.

Quick Facts

Distributive property
(a + b)(c + d) = ac + ad + bc + bd
Every term of the first polynomial multiplies every term of the second (FOIL for two binomials).
Degree rule
deg(A × B) = deg(A) + deg(B)
The product's highest exponent is the sum of the two input degrees.
Max term count
m terms × n terms ≤ m·n terms
Combining like terms after distributing usually shortens the result.

Your Results

Calculated
Product (expanded)
-
A(x) × B(x), like terms combined
Degree of product
-
deg(A) + deg(B)
Leading coefficient
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Coefficient of the highest-power term
Constant term
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Coefficient of the power-0 term

Ready

Enter both polynomials' coefficients, then press Calculate.

How Polynomial Multiplication Works

Multiplying two polynomials means applying the distributive property so that every term in the first polynomial is multiplied by every term in the second polynomial, and then the resulting like terms — terms that share the same power of the variable — are combined by adding their coefficients. For two binomials, this is the familiar FOIL pattern (First, Outer, Inner, Last); for larger polynomials the same distributive logic simply has more terms to track.

Formula and method

If A(x) has terms with coefficients a₀, a₁, … and B(x) has terms with coefficients b₀, b₁, …, each term of A(x) is multiplied by each term of B(x): a term of degree p times a term of degree q produces a term of degree p + q, and its coefficient is the product of the two original coefficients. Once every pairwise product is formed, terms that land on the same power of the variable are added together. This gives two useful rules: the degree of the product always equals the sum of the two input degrees, deg(A × B) = deg(A) + deg(B), and a polynomial with m terms times one with n terms yields at most m × n terms before any combining happens.

Common mistakes

  • Forgetting to distribute every term: each term of the first polynomial must multiply each term of the second — skipping a pairing is the most common FOIL error.
  • Adding exponents incorrectly: when multiplying powers of the same variable, exponents add (x² × x³ = x⁵); they do not multiply.
  • Sign errors: a negative coefficient multiplied by another negative coefficient produces a positive term — track signs term by term rather than at the end.
  • Forgetting to combine like terms: after distributing, terms with the same power of the variable must be added together to reach the simplified final answer.

Checking your result

A fast way to verify a polynomial product is substitution: pick a simple test value such as x = 1, evaluate the two original polynomials at that value and multiply the results, then evaluate the expanded product at the same value. The two numbers must match — if A(1) × B(1) does not equal the expanded product evaluated at 1, a term was dropped, a sign was flipped, or an exponent was added incorrectly somewhere in the expansion.

Applications

Polynomial multiplication underlies expanding factored expressions back into standard form, building area models in algebra (multiplying binomial side lengths to get an area polynomial), combining rate and time expressions in physics and engineering, and multiplying generating functions in combinatorics. In computing, multiplying polynomials is mathematically the same operation as discrete convolution, which shows up in signal processing and in fast multiplication algorithms for large numbers.

Frequently Asked Questions

What is the rule for multiplying two polynomials?
Use the distributive property: multiply every term of the first polynomial by every term of the second polynomial, then combine any like terms by adding the coefficients of terms that share the same power of the variable. For two binomials this is the FOIL pattern (First, Outer, Inner, Last).
How do you find the degree of a product of two polynomials?
Add the degrees of the two polynomials: deg(A × B) = deg(A) + deg(B). For example, multiplying a degree-2 polynomial by a degree-1 polynomial always produces a degree-3 polynomial (assuming neither leading coefficient is zero).
How many terms does the product of two polynomials have?
Before combining like terms, a polynomial with m terms times one with n terms produces at most m × n terms. After combining like terms with the same exponent, the final count is often smaller, and can never exceed one more than the degree of the product.
How do I enter a polynomial that is missing a term, like x³ + 1?
Include a 0 for the missing power. For x³ + 1 (no x² or x term), enter coefficients from the highest degree down as 1, 0, 0, 1, which the calculator reads as 1x³ + 0x² + 0x + 1.

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