How this calculation works
Read both descending coefficient lists, align equal powers for addition/subtraction, or distribute every coefficient pair for multiplication. Rational coefficients remain exact, and the resulting coefficient table retains internal zero powers.
Add matching powers; (AB)ₖ = Σᵢ₊ⱼ₌ₖ aᵢbⱼA worked example
(x² + 2x + 1)(x − 1) = x³ + x² − x − 1. The middle coefficients come from combining every pair with the same total power.
Good to know
Bounded exact rational inputs within ±1,000,000, up to 6 decimal places or integer fraction denominators up to 1,000,000. Large intermediate fractions have a size limit. No arbitrary expression evaluation, multiple variables or general computer-algebra system. Approximate lengths and quadratic square roots are labeled separately. Each input has degree at most 20; a product can have degree at most 40. Coefficient lists only, without variable names or expression syntax.
Reference: OpenStax: polynomial division (opens in a new tab) ↗
A coefficient list needs every power position
The list 1 0 −2 3 represents x³ − 2x + 3. The zero preserves the missing x² position. Omitting it would instead describe x² − 2x + 3, a different polynomial.
Addition and subtraction align powers before combining coefficients. Multiplication distributes all term pairs, then combines pairs with the same total power. For (x² + 2x + 1)(x−1), the x² coefficient becomes 2−1 = 1, while the x coefficient becomes 1−2 = −1.
Fraction arithmetic keeps coefficients such as 1/3 exact instead of replacing them with a short decimal approximation. Decimal inputs are treated as the exact finite decimals entered, within the stated parser bounds.
Degree can decrease through cancellation or increase through multiplication
Subtracting a polynomial from itself gives the zero polynomial. Its degree is not reported as an ordinary zero: a nonzero constant has degree zero, while the zero polynomial has no degree under the convention used here.
Each input may have degree at most 20. Multiplying two degree-20 inputs can produce degree 40, so the output limit differs from the input limit. A leading zero in a list is removed before the actual degree is identified.
Polynomial division reverses a product when a suitable factor is supplied. Its quotient/remainder identity provides a useful independent check on an expansion; evaluating at several points can catch an error but is not, by itself, a proof of equality for unbounded degrees.
A few questions, answered
Can I enter x²+2x+1 directly?
Use its descending coefficient list: 1 2 1. This page does not parse expression notation.
How are unequal list lengths aligned?
By power, with missing higher coefficients treated as zero. The constant is always the final input entry.
Are fractions permitted?
Yes. Use bounded integer fractions such as −2/3, typed with a normal keyboard minus sign.