Polynomial calculator
Enter two coefficient lists to combine like powers or multiply every term. The calculator shows how each list is interpreted before giving the result.
Polynomial result
Enter the coefficients, then select Calculate.
- A(x) as read
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- B(x) as read
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- Result coefficients (highest power first)
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- Operation used
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Decimal results are displayed with up to two decimal places; meaningful smaller nonzero coefficients use scientific notation. Integer calculations are accepted only while every integer remains exactly representable.
How to add, subtract, and multiply polynomials
This calculator handles three defined operations on two polynomials in one variable, x. It is designed for coefficient lists, not for typing or evaluating arbitrary algebraic expressions.
- Write each polynomial as coefficients from its highest power down to the constant term. Separate the numbers with commas.
- Include a 0 wherever a power is missing. For 2x² − 3, enter 2, 0, -3; the middle zero represents the missing x term.
- Choose A + B, A − B, or A × B. Subtraction keeps the order shown, so A − B is generally different from B − A.
- Select Calculate and first confirm the two “as read” expressions. Then use the expanded result or copy the labeled result. Editing any field clears the previous answer.
How descending coefficient lists represent a polynomial
A polynomial such as 4x⁴ − x³ + 5x + 1 contains an implied coefficient for every power down to x⁰. Its complete list is 4, -1, 0, 5, 1. The zero is necessary because there is no x² term. A leading coefficient of 1 or −1 still belongs in the input even though conventional notation writes x or −x without the numeral.
The number of entries determines the starting degree: one entry is a constant, two entries represent ax + b, and eleven entries represent degree 10 down to the constant. Leading zeros are accepted and removed from the displayed expression, while internal and final zeros keep their place. The result includes its own descending list so it can be pasted into another calculation.
- Commas separate coefficients; spaces around them are optional.
- Use a period for decimal input. Fractions such as 1/2 and algebraic text such as x+1 are not parsed.
- The variable is x and exponents are nonnegative whole numbers only.
Adding and subtracting like powers
Addition and subtraction line up equal powers of x. If one polynomial has no term at a power, its coefficient at that power is zero. For A(x) = 2x² − 3 and B(x) = x + 4, addition gives 2x² + x + 1 because the x², x, and constant coefficients become 2 + 0, 0 + 1, and −3 + 4.
For the same inputs, A − B is 2x² − x − 7. The sign of every coefficient in B is subtracted. This order matters: B − A would be a different polynomial. The tool shows the selected operation beside the answer so copied results keep that context.
Multiplying every term and combining the result
Polynomial multiplication distributes every term of A across every term of B. Coefficients multiply and exponents add, then products with the same exponent are combined. For example, (x + 2)(x − 2) produces x² + 0x − 4, displayed conventionally as x² − 4 while the result list remains 1, 0, -4.
A degree-m polynomial times a degree-n polynomial can have degree m + n, so two degree-10 inputs can produce degree 20. This is why the output may be longer than either input. The calculation is finite and local; it does not factor the answer, solve for roots, draw a graph, or infer a variable expression.
Precision, limits, and polynomial division
Each input may contain at most eleven coefficients, corresponding to degree 10. Values must be finite JavaScript numbers. Whole-number inputs are accepted only inside the safe integer range, and an operation is rejected if an integer product or accumulated coefficient would leave that exact range. Decimal arithmetic retains the browser number result and the display uses at most two decimal places; a small nonzero value is shown in scientific notation rather than as a false zero.
Use smaller coefficients if an overflow message appears. Clear removes both coefficient lists; changing the operation or either list invalidates the old result. This page deliberately stops at addition, subtraction, and multiplication. Polynomial long division has different quotient and remainder inputs and remains a separate linked tool.
Frequently asked questions
Why do I need zeros for missing terms?
The position of each number identifies its power. Without the zero in 2, 0, -3, the list 2, -3 would mean 2x − 3 rather than 2x² − 3.
Can the two polynomials have different degrees?
Yes. The shorter list is aligned with the lower powers. Missing higher-power coefficients are treated as zero.
Why did the result omit a zero term?
Standard polynomial notation leaves out terms whose coefficient is zero. The result coefficient list still includes internal zeros so the positions remain unambiguous.
Does this calculator divide or factor polynomials?
No. It performs only addition, subtraction, and multiplication. Use the linked long-division tool for a quotient and remainder.
Are decimals supported?
Yes, finite decimal coefficients are supported. The display is rounded to at most two decimal places, while very small nonzero coefficients use scientific notation.