Multiplying Polynomials Calculator
Multiply two single-variable polynomials, combine like powers automatically, inspect every coefficient contribution, and export the current calculation as a validated Excel workbook.
Ready for the startup example.
Polynomial coefficients
Choose each declared degree from 1 through 6. Enter ordinary decimal coefficients; commas are accepted only as thousands separators. A blank coefficient inside an otherwise entered polynomial is treated as zero.
Polynomial P(x)
P(x) = x⁴ – 3x² + 2x + 4
Polynomial Q(x)
Q(x) = – 0.5x² + x – 2
Product P(x) × Q(x)
– 0.5x⁶ + x⁵ – 0.5x⁴ – 4x³ + 6x² – 8
The product is a degree 6 polynomial with 6 nonzero terms.
Coefficient convolution
Each row collects all coefficient products whose exponents add to the same output power. Their sum is the coefficient of that power in the final polynomial.
| Power | Contributions | Result coefficient |
|---|---|---|
| x⁶ | (1 × – 0.5) | – 0.5 |
| x⁵ | (1 × 1) + (0 × – 0.5) | 1 |
| x⁴ | (1 × – 2) + (0 × 1) + ( – 3 × – 0.5) | – 0.5 |
| x³ | (0 × – 2) + ( – 3 × 1) + (2 × – 0.5) | – 4 |
| x² | ( – 3 × – 2) + (2 × 1) + (4 × – 0.5) | 6 |
| x¹ | (2 × – 2) + (4 × 1) | 0 |
| x⁰ | (4 × – 2) | – 8 |
Zero coefficients remain in the table because they verify cancellations and missing powers. The simplified expression omits those zero terms.
How to use the multiplying polynomials calculator
What this calculator does
This calculator multiplies two polynomials in one variable, written as P(x) and Q(x). It applies the distributive property to every pair of terms and combines products that have the same exponent. The result is an exact algebraic identity for the entered decimal coefficients, subject only to ordinary floating-point display rounding for very long decimals. It does not factor the answer, find roots, graph the functions, or evaluate the product at a particular x-value. Use it when you need the expanded, simplified coefficient form.
When to use it
It is useful for checking homework, expanding a product before differentiation or integration, verifying a symbolic derivation, or preparing coefficient data for a spreadsheet or numerical model. It also helps reveal cancellations: a power can disappear from the simplified expression when positive and negative contributions sum to zero.
How to calculate
- The calculator opens with a complete demonstration: P(x) = x⁴ – 3x² + 2x + 4 and Q(x) = – 0.5x² + x – 2. The result and a validated example Excel workbook are available immediately.
- Choose Degree of P(x) and Degree of Q(x). A degree is the greatest exponent you want to enter, from 1 through 6. Higher choices reveal more coefficient fields.
- Replace the visible a-coefficients and b-coefficients. Enter a signed integer or decimal such as 3, – 0.5, or 1,250.75. A blank field inside an otherwise entered polynomial means zero; commas must be conventional thousands separators, not decimal commas.
- Read Product P(x) × Q(x) and the supporting metrics. Then inspect the Coefficient convolution table to see which term products produced each power.
- Select Download Excel to export the current inputs, summary, result coefficients, and term-by-term contributions as a real .xlsx workbook. Reset clears the demonstration coefficients, returns the degree selectors to 3 and 2, removes calculated content, and disables export until both polynomials contain a complete valid state again.
Input guide
Degree of P(x) and Degree of Q(x) are required whole-number selections from 1 to 6. They determine which coefficient fields are active and the maximum possible product degree. For example, degrees 4 and 2 permit a result up to degree 6. Selecting a higher degree adds fields but does not by itself make the polynomial higher-degree if the new leading coefficient is zero. A common mistake is to confuse the number of terms with the degree.
Coefficient a₆ through a₀ define P(x), while Coefficient b₆ through b₀ define Q(x). Only coefficients up to the selected degree are used. Each visible field accepts a finite signed decimal in U.S. notation, with an absolute value no greater than 10¹⁰⁰. At least one field in each polynomial must be explicitly entered; after that, blanks count as zero. Increasing a coefficient changes every output coefficient paired with it. Negative values reverse the signs of their contributions. Do not enter x, powers, fractions such as 1/2, decimal commas such as 1,5, or scientific notation such as 1e3; enter the numerical coefficient only.
Output guide
Product P(x) × Q(x) is the simplified expanded polynomial. Product degree is the highest power with a nonzero resulting coefficient; the zero polynomial has no defined degree and is shown as “Undefined.” Nonzero terms counts the powers that remain after like terms are combined. Leading coefficient is the coefficient attached to the product's highest nonzero power, and Constant term is the coefficient of x⁰. Coefficient pairs counts all active term pairings, including pairings with zero coefficients, so it equals (declared degree of P + 1) × (declared degree of Q + 1).
The Coefficient convolution table has three columns. Power identifies the output exponent. Contributions lists every aᵢ × bⱼ product for which i + j equals that exponent. Result coefficient is their sum. A zero result can mean all contributions were zero or that nonzero contributions canceled. The table is an exact structural explanation of the multiplication, not a recommendation or approximation.
Worked example
With the startup values, the x⁴ term of P(x) has coefficient 1 and the x² term of Q(x) has coefficient – 0.5, so the highest product term is (1 × – 0.5)x⁶ = – 0.5x⁶. For x⁴, the relevant products are 1 × – 2, 0 × 1, and – 3 × – 0.5. Their sum is – 2 + 0 + 1.5 = – 0.5. Repeating this for every exponent gives – 0.5x⁶ + x⁵ – 0.5x⁴ – 4x³ + 6x² + 0x – 8, and the simplified display omits the zero x term.
Learn more
The method is repeated distribution followed by combining like terms. The OpenStax College Algebra section on polynomials gives the general distributive procedure and explains why powers with the same exponent are combined. For additional worked examples, see OpenStax's lesson on multiplying polynomials. A complementary presentation of the vertical organization method appears in LibreTexts' multiplying polynomial expressions guide.
Why convolution is the right model
Each term aᵢxⁱ multiplied by bⱼxʲ becomes aᵢbⱼxⁱ⁺ʲ because coefficients multiply and exponents add. Terms with equal exponent i + j are like terms, so their coefficients add. This coefficient-by-coefficient rule is called discrete convolution. The calculator uses the same canonical result data for the expression, metrics, table, accessible summary, and workbook, preventing one view from drifting away from another.
Common mistakes and interpretation limits
- Do not add exponents across a sum. Exponents add only when multiplying powers of the same variable.
- FOIL is only a memory aid for two binomials. Distribution or convolution works for every supported degree.
- A missing x-power has coefficient zero. Entering 0 explicitly can make the table easier to audit.
- If a leading coefficient is zero, the actual polynomial degree is lower than the selected declared degree.
- Very large coefficients may produce long scientific-notation displays. The calculator rejects nonfinite or overflowed results rather than showing NaN or Infinity.
For classroom work, compare the table's contribution row with your handwritten distribution. If the final expression differs, the first mismatching power usually identifies a sign error, a skipped term, or an exponent that was added incorrectly.