Polynomial Calculator

What Is a Polynomial?

A polynomial is an expression built from one or more terms, where each term is a coefficient multiplied by a variable raised to a whole-number power (or a plain number by itself). For example, in 3x² + 2x − 5, there are three terms: 3x², 2x, and −5.

• Coefficient — the number multiplying the variable, such as the 3 in 3x² or the 2 in 2x. A term with no visible number, like x² by itself, has an implied coefficient of 1.
• Exponent (power) — the whole number the variable is raised to, such as the 2 in x². A term with no exponent shown, like 2x, is understood to be raised to the first power.
• Constant term — a term with no variable at all, like −5. It can be thought of as that number multiplied by x⁰, since anything raised to the power of 0 equals 1.
• Degree — the highest exponent appearing anywhere in the polynomial. In 3x² + 2x − 5, the highest power is 2, so the polynomial has degree 2 (also called a quadratic). A degree-1 polynomial like 2x + 7 is linear, a degree-3 polynomial is cubic, and so on.

Polynomials are normally written in standard form, meaning the terms are ordered from the highest power down to the lowest (the constant term last). This calculator both reads and outputs polynomials in standard form.

Adding and Subtracting Polynomials

Adding or subtracting two polynomials comes down to one rule: you can only combine like terms — terms that have the exact same variable raised to the exact same power. You add or subtract their coefficients and keep the variable part unchanged. Terms with different powers, like 3x² and 2x, can never be combined; they simply sit side by side in the result.

Worked example (addition): Add (2x² + 3x − 1) + (x² − 4x + 5).

• x² terms: 2x² + 1x² = 3x²
• x terms: 3x + (−4x) = −1x, written as −x
• Constants: −1 + 5 = 4

Combining those gives a final answer of 3x² − x + 4. Subtraction works the same way, except every term in the second polynomial has its sign flipped before combining — subtracting (x² − 4x + 5) from (2x² + 3x − 1) means combining 2x² − x², 3x − (−4x), and −1 − 5, giving x² + 7x − 6.

Multiplying Polynomials

Multiplying polynomials uses the distributive property: every term in the first polynomial gets multiplied by every term in the second polynomial, and then the resulting products are combined by adding together any like terms.

Worked example (FOIL): for two binomials, (x + 2)(x + 3), this distribution has a well-known shortcut called FOIL — First, Outer, Inner, Last:

• First: x × x = x²
• Outer: x × 3 = 3x
• Inner: 2 × x = 2x
• Last: 2 × 3 = 6

Adding those four products together gives x² + 3x + 2x + 6, and combining the like x terms gives a final answer of x² + 5x + 6.

FOIL is really just a memory trick for the general distributive rule, and it only works because a binomial has exactly two terms (so there are exactly four pairings: First-First, First-Second, Second-First, Second-Second). The moment either polynomial has three or more terms — a trinomial times a binomial, or a trinomial times a trinomial — there are more than four pairings to track, so FOIL no longer applies by name. The underlying idea generalizes perfectly, though: distribute every term of the first polynomial across every term of the second, then combine like terms, exactly as this calculator's Multiply button shows step by step regardless of how many terms either polynomial has.

Evaluating a Polynomial at a Value

Evaluating a polynomial means substituting a specific number in for x everywhere it appears, then simplifying using the normal order of operations (exponents before multiplication, multiplication before addition/subtraction) to get a single numeric answer.

For example, to evaluate 3x² + 2x − 5 at x = 2, substitute 2 for every x:

3(2)² + 2(2) − 5 = 12 + 4 − 5 = 11

Each term is worked out on its own first — squaring or raising to whatever power comes first, then multiplying by the coefficient — and only after every term has been reduced to a plain number are they added and subtracted together to reach the final result.

Frequently Asked Questions

What is the degree of a polynomial?

The degree is the highest exponent that appears on the variable anywhere in the polynomial. A constant like 7 has degree 0, a linear expression like 2x + 1 has degree 1, and an expression like 3x² + 2x − 5 has degree 2 — commonly called a quadratic. If you're specifically working with a quadratic and need its roots rather than general arithmetic on it, the quadratic formula calculator is built exactly for that.

What's the correct format to type a polynomial into this calculator?

Use standard notation with a caret for exponents, such as 3x^2 + 2x - 5 or 2x^3 - x + 4. Spaces are optional, coefficients of 1 can be left off (so x^2 is read as 1x^2), and any power you don't mention is simply treated as having a coefficient of 0. Exponents up to x^6 are supported; anything higher will produce an error message telling you which term needs to be lowered.

Can I use this for polynomials with more than one variable?

No — this calculator only supports a single variable, x. Multi-variable polynomials (like 2x²y + 3xy − y²) introduce an entirely different set of rules for what counts as a "like term," since two terms would need matching exponents on every variable, not just one, before they could be combined. Keeping this tool single-variable keeps the input format simple and unambiguous, and covers the overwhelming majority of algebra and precalculus coursework, which is almost always framed in terms of a single variable x.

What does it mean for two terms to be "like terms"?

Two terms are "like terms" if they have the exact same variable raised to the exact same power — only their coefficients are allowed to differ. For example, 3x² and −7x² are like terms (both are "x squared" terms) and can be combined into −4x², but 3x² and 3x are not like terms, even though they share a coefficient, because their exponents (2 and 1) don't match. Only like terms can ever be added or subtracted together.

How is multiplying polynomials related to FOIL?

FOIL (First, Outer, Inner, Last) is a memorization shortcut for multiplying exactly two binomials — it's just a way of making sure you distribute every term of the first binomial across every term of the second without missing a pairing. It's a special case of the more general distributive rule that this calculator's Multiply button applies to polynomials of any size: every term in Polynomial A is multiplied by every term in Polynomial B, and the resulting products are then combined wherever they share the same power of x.

Why does my result show a higher degree than either polynomial I entered?

This is expected when multiplying. Adding or subtracting two polynomials never raises the degree beyond whichever input already had the higher one, but multiplying does — the degree of a product is the sum of the two input degrees. For example, multiplying two degree-2 (quadratic) polynomials together always produces a degree-4 result, since the highest-power terms from each side multiply together (x² × x² = x⁴).