Polynomial Multiplication Calculator

About the Polynomial Multiplication Calculator

Multiplying polynomials, often called expanding brackets, is one of the core skills of algebra. It is how you go from a factored form such as (2x + 3)(x − 4) to the standard form 2x² − 5x − 12, and it appears in solving equations, simplifying expressions, finding areas and volumes, and working with probability generating functions and series.

This polynomial multiplication calculator multiplies two or three polynomials in x. It lists every term-by-term product before like terms are combined, names the FOIL products for two binomials, gives the expanded result in standard form, and reports its degree and number of terms. Coefficients are kept as exact fractions, so there is no rounding.

How to Use the Polynomial Multiplication Calculator

Enter the first polynomial, using x as the variable and ^ for powers, such as 2x + 3 or x^2 - 3x + 1.

Enter the second polynomial.

Optionally enter a third polynomial to multiply all three.

You can write products inside a factor, such as (x + 1)^2, and the calculator expands them too.

The Formulas

  (a_m x^m + … + a_0)(b_n x^n + … + b_0) = Σ a_i b_j x^(i + j)
  degree of the product = degree of the first + degree of the second
  terms before combining = terms in the first × terms in the second

Step-by-Step Example

Multiply (2x + 3)(x − 4).

  First:  2x × x    = 2x²
  Outer:  2x × (−4) = −8x
  Inner:  3 × x     = 3x
  Last:   3 × (−4)  = −12
  combine like terms: 2x² + (−8x + 3x) − 12 = 2x² − 5x − 12

Multiply (x² + x + 1)(x − 1)(x + 1). The first two factors give six products: x³, −x², x², −x, x and −1, which combine to x³ − 1. Multiplying x³ − 1 by x + 1 gives x⁴ + x³ − x − 1, a degree 4 polynomial with four terms.

The Distributive Property

Every polynomial multiplication rests on the distributive property: a(b + c) = ab + ac. Multiplying (2x + 3)(x − 4) distributes 2x over (x − 4) and then 3 over (x − 4). Writing the products in a grid, with the terms of one factor across the top and the other down the side, is a tidy way to make sure no product is missed, especially with trinomials.

FOIL is simply a memory aid for two binomials: first, outer, inner, last. It does not extend to longer polynomials, where you multiply every term of one factor by every term of the other. A trinomial times a trinomial gives nine products before combining.

Special Products Worth Knowing

Some products appear so often that it pays to recognise them:

  (a + b)² = a² + 2ab + b²
  (a − b)² = a² − 2ab + b²
  (a + b)(a − b) = a² − b²
  (a + b)³ = a³ + 3a²b + 3ab² + b³
  (a − b)(a² + ab + b²) = a³ − b³

For instance, (x + 1)³ expands to x³ + 3x² + 3x + 1, and (x − 1)(x² + x + 1) = x³ − 1. The calculator confirms these patterns and handles any coefficients, including fractions and negatives.

Degree and Number of Terms

The degree of a product is the sum of the degrees of the factors, because the highest powers multiply together and their coefficients cannot cancel. The number of terms is harder to predict: it is at most the product of the term counts, but like terms often combine, and sometimes cancel completely, as the middle terms do in (a + b)(a − b) = a² − b². The calculator reports both figures so you can check your expansion has the right shape.

Exact Fractions

Decimal and fractional coefficients are converted to exact fractions, so (x/2 + 1)(x/3 − 1) gives (1/6)x² − (1/6)x − 1 exactly, with no rounding drift. This matters when the result feeds into further steps, such as factoring or comparing coefficients.

Multiplying Polynomials in Real Problems

Expanding products comes up whenever quantities multiply. A rectangle with sides x + 3 and 2x − 1 has area (x + 3)(2x − 1) = 2x² + 5x − 3. A box made by cutting squares of side x from the corners of a 20 by 30 sheet has volume x(20 − 2x)(30 − 2x), which expands to 4x³ − 100x² + 600x, ready for finding its maximum with calculus. Compound growth over two periods at rates r and s multiplies (1 + r)(1 + s) = 1 + r + s + rs, showing the small extra rs term that interest on interest adds.

In probability, multiplying generating functions counts outcomes. The number of ways to roll each total with two dice is given by the coefficients of (x + x² + x³ + x⁴ + x⁵ + x⁶)², where the coefficient of x⁷ is 6, the six ways to roll a seven.

Checking an Expansion

A quick check is to substitute a number. If (2x + 3)(x − 4) = 2x² − 5x − 12, then at x = 1 both sides must agree: (5)(−3) = −15 and 2 − 5 − 12 = −15. One matching value does not prove the expansion right, but a mismatch proves it wrong, and checking x = 0 confirms the constant term, which is always the product of the constant terms of the factors.

Understanding Your Result

The headline gives the expanded product in standard form, highest power first.

The problem line restates the factors.

The degree line gives the degree, the number of terms and the sum of the factor degrees.

The method line lists the FOIL products for two binomials, or the number of term-by-term products.

The steps show the products and the combined result after each multiplication.

When Should You Use This Calculator?

Use it to check expansions in algebra homework.

Use it to expand a factored polynomial back into standard form.

Use it to verify special product patterns.

Common Mistakes

Missing a product. Every term must multiply every term of the other factor.

Sign errors. A negative times a negative is positive: (−4)(−3) = 12.

Squaring a sum term by term. (a + b)² is not a² + b²; the middle term 2ab is needed.

Adding exponents wrongly. x² × x³ = x⁵, not x⁶.