Polynomial Multiplication Calculator
Multiply two polynomial expressions and get a detailed step-by-step solution showing every term-product pair, the expanded unsimplified form, like-term grouping, and the final simplified polynomial. Supports binomials, trinomials, and higher-degree polynomials.
🔢 Multiply Polynomials
Enter two single-variable polynomials using standard algebraic notation (e.g., x^2 + 3x + 2 or 3x - 5).
x^2 + 3x + 2 (use ^ for exponents)📖 Deep-Dive Reference Guide: How to Multiply Polynomials
Looking for complete algebraic explanations, FOIL vs. distributive property rules, box method grids, exponent rules, and worked monomial, binomial, and trinomial examples? Read our comprehensive master guide.
Read Full Polynomial Multiplication Guide & Examples →How to Use the Polynomial Multiplication Calculator
- Enter your first polynomial in the "First Polynomial (A)" field using standard notation: use x^2 for x², separate terms with + or −, and enter coefficients directly before the variable (e.g., 3x^2 - 2x + 1).
- Enter your second polynomial in the "Second Polynomial (B)" field.
- Click a Quick Example chip to auto-fill a demonstration expression pair.
- Click "Calculate" (or press Enter) to view your full step-by-step solution.
- Use "⇅ Swap" to exchange Polynomial A and B without re-typing.
- Click "📋 Copy Result" to copy the final answer to your clipboard.
- Click "More About This Result" to view the degree, leading coefficient, constant term, and number of terms.
- Click "Reset" to clear all fields and start fresh.
How Polynomial Multiplication Is Calculated
Understanding Polynomial Multiplication, FOIL & Like Terms
Polynomial multiplication is a fundamental algebraic operation built on the distributive property. For two polynomials A(x) and B(x), the product is found by multiplying each term of A by each term of B — producing a set of monomial products — then combining all terms with matching exponents. When both expressions are binomials (two terms each), the FOIL method (First, Outside, Inside, Last) provides a convenient shortcut that identifies the same four products. For larger polynomials with three or more terms, FOIL does not apply and the full distributive method must be used. A key insight: when multiplying monomials, coefficients are multiplied and exponents are added (xᵃ · xᵇ = x^(a+b)); when combining like terms, exponents stay the same and only coefficients are added.
Published Clinical & Scientific References
- Stewart, Redlin & Watson — Algebra and Trigonometry (Polynomial Operations).
- Khan Academy — Multiplying Polynomials (Distributive Property & FOIL).
Frequently Asked Questions
What is a polynomial multiplication calculator?
A polynomial multiplication calculator is an online tool that automatically multiplies two polynomial expressions and shows the expanded, simplified result along with detailed step-by-step working so you can learn and verify the process.
How do you multiply two polynomials?
Use the distributive property: multiply every term in the first polynomial by every term in the second polynomial, collect all the resulting products, then combine like terms (those with equal exponents). Sort the final result in descending exponent order.
What is the FOIL method?
FOIL stands for First, Outside, Inside, Last. It is a memory aid for multiplying exactly two binomials. FOIL does not apply to trinomials or larger polynomials — use the full distributive method instead.
Can this calculator multiply trinomials?
Yes. This calculator supports polynomials of any degree and any number of terms, including trinomials, as long as the expressions use one variable (x) and integer or decimal coefficients.
Can I multiply polynomials with negative coefficients?
Yes. Negative coefficients are fully supported. The parser correctly handles leading negatives (e.g., −x² + 3x) and negative terms anywhere in the expression.
What happens to exponents when multiplying polynomial terms?
When multiplying two terms, you multiply the coefficients and add the exponents. For example: (3x²)(2x³) = 6x⁵. This follows the exponent product rule: xᵃ · xᵇ = x^(a+b).
How are like terms combined after multiplication?
Like terms share the same variable exponent. After expanding the multiplication, all products with identical exponents are grouped, and their coefficients are added. For example: 3x² + 5x² = 8x².
Does this calculator show step-by-step working?
Yes. After clicking Calculate, the tool shows four steps: (1) every individual term-product pair with color-coded source tracking, (2) the full expanded unsimplified expression, (3) like-terms grouped by exponent with combined coefficients, and (4) the final simplified polynomial.
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How to Multiply Polynomials: FOIL, Box Method & Examples
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