# Multiplying Polynomials Calculator

URL: https://speedycalculator.com/calculators/math/multiplying-polynomials/
Last updated: 2026-10-30

> The multiplying polynomials calculator multiplies two polynomials of degree up to 6 and returns the product in standard form.

## Inputs

- Degree of P(x)
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀
- Degree of Q(x)
- b₆ (x⁶)
- b₅ (x⁵)
- b₄ (x⁴)
- b₃ (x³)
- b₂ (x²)
- b₁ (x)
- b₀

## Results

- P(x) × Q(x)
- P(x)
- Q(x)
- Degree of the product

## Formula

```
(Σ aᵢxⁱ)(Σ bⱼxʲ) = Σ (aᵢbⱼ) x^(i+j)
```

## Explanation

Multiplying polynomials is the distributive property applied many times: every term of the first polynomial multiplies every term of the second, and like terms are combined. The degree of the product is the sum of the degrees.

### How to use the Multiplying Polynomials Calculator

- Choose the **degree** of each polynomial – the matching coefficient fields appear.
- Enter the **coefficients** from the highest power down; leave a field at 0 for a missing power.
- Read the result, written with powers of x in descending order.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator convolves the coefficient lists – the coefficient of xⁿ in the product is the sum of aᵢbⱼ over all i + j = n.

## Worked example

Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): 2; a₁ (x): 0; a₀: -1; Degree of Q(x): 2; b₂ (x²): 1; b₁ (x): -3; b₀: 2.

Results — **P(x) × Q(x) = x⁵ − x⁴ − 4x³ + 3x² + 3x − 2**; P(x) = x³ + 2x² − 1; Q(x) = x² − 3x + 2; Degree of the product = 5.

## Key terms

- **Distributive property**: a(b + c) = ab + ac.

## FAQ

**What is the degree of a product?**
The sum of the degrees: a cubic times a quadratic is degree 5.

**How many terms before simplifying?**
The number of terms of P times the number of terms of Q.

**Is polynomial multiplication commutative?**
Yes, P·Q = Q·P.

**What is the fastest method by hand?**
Use a grid (table) or vertical multiplication like long multiplication.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)