Multiplying Polynomials
Multiply two quadratic polynomials (up to degree 2) and get the expanded result showing coefficients for all terms up to x^4. Free online polynomial multiplication tool.
Polynomial A (a₀ + a₁x + a₂x²)
Polynomial B (b₀ + b₁x + b₂x²)
About This Calculator
The Multiplying Polynomials calculator computes the product of two polynomials up to degree 2 (quadratic). Enter coefficients for each polynomial in the form a0 + a1x + a2x^2, and the tool returns the expanded result with all coefficients displayed. This free online tool is ideal for algebra students learning polynomial multiplication, teachers preparing examples, and anyone who needs to expand products of quadratic expressions quickly.
Multiplication of polynomials follows the distributive law (the same principle used in the FOIL method for binomials): every term in the first polynomial is multiplied by every term in the second, and like terms are combined. For example, (a0 + a1x + a2x^2)(b0 + b1x + b2x^2) = a0b0 + (a0b1 + a1b0)x + (a0b2 + a1b1 + a2b0)x^2 + (a1b2 + a2b1)x^3 + a2b2x^4. For two quadratics, the resulting polynomial can have terms up to degree 4 (x^4). The result section shows the full expanded polynomial and a breakdown table of each coefficient from constant up to x^4.
Examples: (1 + 2x + 3x^2)(4 + 5x + 6x^2) = 4 + 13x + 28x^2 + 27x^3 + 18x^4. (x^2 - 1)(x^2 + 1) = x^4 - 1. (2x + 1)(x^2 - x + 3) = 2x^3 - x^2 + 5x + 3. Enter any zero coefficients for missing terms -- the calculator handles partial polynomials correctly.
Who this is for: High school and college students studying algebra, precalculus, or polynomial functions; math teachers creating practice problems; tutors demonstrating polynomial expansion; and professionals who need quick polynomial product verification. The calculator supports real-number coefficients including integers, fractions, decimals, and negative values.
Frequently Asked Questions
How does polynomial multiplication work?
Polynomial multiplication uses the distributive property: each term of the first polynomial is multiplied by every term of the second polynomial. The result is the sum of all these products with like terms combined. For degree-2 polynomials, the result is degree 4.
What is the highest degree polynomial I can multiply?
Each input polynomial can have terms up to degree 2 (quadratic). The product of two quadratics can reach degree 4 (quartic). For multiplying binomials (degree 1), use the FOIL or Multiplying Binomials calculators.
Can I multiply polynomials with missing terms?
Yes, simply enter 0 for the missing term coefficients. For example, to multiply (x^2 + 5)(2x - 3), enter A coefficients: 5 (constant), 0 (x), 1 (x^2); B coefficients: -3 (constant), 2 (x), 0 (x^2).
Is this polynomial multiplication tool free?
Yes, all calculators on Calculy are completely free to use with no registration or downloads required.
What types of coefficients are supported?
The calculator supports all real number coefficients including integers, fractions, decimals, and negative numbers. Results are shown with proper sign formatting.
Can I share my polynomial multiplication result?
Yes, use the Share button to copy a URL containing your current inputs. Anyone opening that link will see the same polynomials and product results automatically.