Math & Business Productivity

Polynomial Multiplication Calculator

Free polynomial multiplication calculator. Multiply binomials (ax + b)(cx + d) instantly using FOIL expansion with roots and quadratic term analysis.

Reviewed by Dr. Marcus Vance, Ph.D. (Ph.D. Applied Mathematics (MIT))

Verified for accuracy•Updated 2026

Interactive Calculator

1. Provide Details

Coeff a in (ax + b)

Constant b in (ax + b)

Coeff c in (cx + d)

Constant d in (cx + d)

2. Output Results

Expanded Polynomial (Standard Form)8x² + 2x - 15

Quadratic Term (x²)8x²

Linear Term (x)2x

Constant Term-15

Polynomial Zeros / Rootsx₁ = -1.5000, x₂ = 1.2500

How to Calculate: Formula & Steps

Applies the FOIL (First, Outside, Inside, Last) multiplication algorithm to expand two binomial expressions.

Formula Used:(ax + b)(cx + d) = (a·c)x² + (a·d + b·c)x + (b·d)

Step-by-Step Calculation Example

For (2x + 3)(4x - 5): First = 8x², Outside = -10x, Inside = 12x, Last = -15. Combined: 8x² + 2x - 15.

Common Mistakes to Avoid

Forgetting to distribute negative signs during multiplication.
Attempting to add terms with different powers (e.g., x² + x).

Practical Use Cases

Algebra 1 & Algebra 2 polynomial expansion
Quadratics factoring and root verification
Engineering curve model expansions

Expert Tips

Always write the expanded polynomial in standard descending degree order.

Disclaimer

This calculation tool is provided for educational and informational estimation purposes only. Results are based on mathematical formulas and user-supplied parameters. They do not constitute formal underwriting, financial, tax, engineering, or legal determinations.

Frequently Asked Questions About Polynomial Multiplication Calculator

How do you multiply two binomials using FOIL?

FOIL stands for First, Outside, Inside, Last. Multiply the First terms of each binomial, then the Outside terms, the Inside terms, and the Last terms, then combine like terms.

How do I find the zeros or roots of a polynomial?

Set each factor equal to zero: ax + b = 0 => x = -b/a, and cx + d = 0 => x = -d/c.