Multiplying Polynomials Calculator
Multiply two polynomials and get the expanded and simplified result. Includes optional step-by-step.
Background
Polynomial multiplication is repeated distribution: each term in the first polynomial multiplies each term in the second, then you combine like terms.
How to use this calculator
- Enter the two polynomials in Polynomial A and Polynomial B.
- Click Calculate to get expanded and simplified results.
How this calculator works
- Parse each polynomial into terms like ax^n.
- Distribute: multiply A by each term in B.
- Combine like terms (same exponent) to simplify.
Formula & Equation Used
Polynomial multiplication uses the distributive property:
(a + b)(c + d) = ac + ad + bc + bd
More generally, multiply every term in the first polynomial by every term in the second polynomial:
(aₘxᵐ + … + a₁x + a₀)(bₙxⁿ + … + b₁x + b₀)
Then add all term-by-term products and combine like terms.
Example Problems & Step-by-Step Solutions
Example 1 — Binomial × Binomial
Multiply (x + 2)(x + 5).
Step-by-step
- Distribute: (x + 2)(x + 5) = x(x + 5) + 2(x + 5)
- Multiply: = (x^2 + 5x) + (2x + 10)
- Combine like terms: x^2 + 7x + 10
Answer: x^2 + 7x + 10
Tip: This is the classic FOIL case.
Example 2 — Trinomial × Binomial
Multiply (x^2 + 2x + 1)(x − 3).
Step-by-step
- Distribute x: x(x^2 + 2x + 1) = x^3 + 2x^2 + x
- Distribute −3: −3(x^2 + 2x + 1) = −3x^2 − 6x − 3
- Add results: x^3 − x^2 − 5x − 3
Answer: x^3 − x^2 − 5x − 3
Tip: A 3-term polynomial times a 2-term polynomial creates 6 products before combining.
Example 3 — Difference of Squares
Multiply (3x − 1)(3x + 1).
Step-by-step
- Distribute: 3x(3x + 1) − 1(3x + 1)
- Multiply: = (9x^2 + 3x) + (−3x − 1)
- Combine like terms: 9x^2 − 1
Answer: 9x^2 − 1
Tip: This follows (a − b)(a + b) = a^2 − b^2.
Frequently Asked Questions
Q: Do I need to type parentheses?
No. Parentheses are optional and ignored. Just enter each polynomial expression.
Q: What inputs are supported?
Single-variable polynomials with +/− signs, integer/decimal coefficients, and exponents using ^.
Q: Can I type something like x(x+2) in one box?
Not in this version. Enter each polynomial as a sum of terms in its own box (example: type x and x+2 separately).
Q: Why am I getting a “Could not parse part” warning?
It usually means an unsupported format (like x2 instead of x^2, or extra symbols). Rewrite using ^ for powers and normal +/− signs.
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