Multiplying Polynomials Calculator
Enter the degree and coefficients of two polynomials P(x) and Q(x), then press Calculate. The tool multiplies every term of P by every term of Q using the distributive property, collects like terms, and displays the expanded product in standard form. Each multiplication step is shown so you can follow the working.
How to multiply polynomials
To multiply two polynomials, use the distributive property: every term of the first polynomial is multiplied by every term of the second polynomial, then all partial products that share the same power of x are added together. For a polynomial with m terms and one with n terms, you generate m x n partial products before combining like terms. The result is always a polynomial whose degree equals the sum of the degrees of the two factors. For example, multiplying a quadratic (degree 2) by a cubic (degree 3) always gives a degree-5 polynomial.
The FOIL method for binomials
When both polynomials have exactly two terms (binomials), the distributive process has a memorable shortcut called FOIL: First, Outer, Inner, Last. For (ax + b)(cx + d): First gives acx^2, Outer gives adx, Inner gives bcx, and Last gives bd. Adding them and combining the two middle terms gives acx^2 + (ad + bc)x + bd. FOIL is simply the distributive property applied to two-term polynomials and produces the same result as the general method. For polynomials with three or more terms, the general distributive approach must be used instead.
Degree, leading coefficient, and constant term
Three properties of the product can be read off without expanding fully. The degree of P(x) x Q(x) is always deg(P) + deg(Q). The leading coefficient of the product is the product of the two leading coefficients. The constant term of the product is the product of the two constant terms (the values when x = 0). Knowing these three quantities lets you check your work quickly: if the degree or leading coefficient of your expanded result does not match, there is a multiplication error somewhere.
Special products and factoring connection
Certain polynomial products appear so often that their patterns are worth memorizing: the square of a sum (a + b)^2 = a^2 + 2ab + b^2, the difference of squares (a + b)(a - b) = a^2 - b^2, and the sum and difference of cubes. These identities work in reverse as factoring formulas. Recognizing that x^2 - 9 = (x + 3)(x - 3), for example, allows fast factoring without the quadratic formula. The reference table above lists the most common identities.
Common polynomial product identities
| Identity name | Form | Expanded product |
|---|---|---|
| Square of a sum | (a + b)^2 | a^2 + 2ab + b^2 |
| Square of a difference | (a - b)^2 | a^2 - 2ab + b^2 |
| Difference of squares | (a + b)(a - b) | a^2 - b^2 |
| Cube of a sum | (a + b)^3 | a^3 + 3a^2 b + 3ab^2 + b^3 |
| Cube of a difference | (a - b)^3 | a^3 - 3a^2 b + 3ab^2 - b^3 |
| Product of sum and square | (a + b)(a^2 - ab + b^2) | a^3 + b^3 |
| Product of difference and square | (a - b)(a^2 + ab + b^2) | a^3 - b^3 |
| General FOIL | (ax + b)(cx + d) | acx^2 + (ad + bc)x + bd |
These identities are special cases of polynomial multiplication that appear frequently in algebra.
Frequently asked questions
What is the degree of the product of two polynomials?
The degree of the product is always equal to the sum of the degrees of the two factor polynomials, provided neither polynomial is identically zero. For instance, a degree-3 polynomial times a degree-4 polynomial always produces a degree-7 polynomial. This follows directly from the rule of exponents: multiplying x^a by x^b gives x^(a+b), so the highest-degree term of the product comes from multiplying the leading terms of each factor.
How does the FOIL method work, and when can I use it?
FOIL stands for First, Outer, Inner, Last, and it is a memory aid for multiplying two binomials (two-term polynomials). For (ax + b)(cx + d), you multiply First terms (a x c = ac), Outer terms (a x d = ad), Inner terms (b x c = bc), and Last terms (b x d = bd), then write the result as acx^2 + (ad + bc)x + bd. FOIL only applies to binomial by binomial products. For trinomials or higher-degree polynomials, you must use the full distributive property: multiply every term of the first polynomial by every term of the second.
Do the polynomials need to have the same degree?
No. You can multiply polynomials of any degrees - a linear polynomial by a degree-6 polynomial, a quadratic by another quadratic, or any other combination. This calculator supports degrees from 1 to 6 for each factor, so the product can be up to degree 12. Coefficients can be any real number including negative values and decimals.
What is the constant term of the product?
The constant term (the term with no x) in the product equals the product of the constant terms of the two factor polynomials. For P(x) x Q(x), the constant term is P(0) x Q(0), because setting x = 0 zeroes out every term that contains x, leaving only the constant parts. This is a useful quick check: if your expanded product has the wrong constant term, at least one multiplication step was done incorrectly.
How many partial products are generated when multiplying two polynomials?
The number of partial products (before combining like terms) equals the number of terms in the first polynomial multiplied by the number of terms in the second polynomial. A trinomial (3 terms) times a quadratic with 3 terms generates 9 partial products. Some of those partial products may share the same exponent and get combined into a single term in the final result, so the final answer often has fewer terms than the number of partial products.
What is a polynomial?
A polynomial is an algebraic expression made up of one or more terms, where each term is a product of a constant (the coefficient) and a non-negative integer power of a variable (usually x). Examples: 3x^2 - 2x + 1 is a quadratic polynomial (degree 2), and 5x^3 + x is a cubic polynomial (degree 3). Terms with the same power of x are called like terms and can be combined by adding their coefficients.