Polynomial Calculator

A polynomial calculator takes two expressions like 2x^2 + 3x - 5 and x^2 - x + 4 and returns their sum, difference, or product in standard form. Adding the two above gives 3x² + 2x - 1; subtracting gives x² + 4x - 9; multiplying gives 2x⁴ + x³ + 0x² + 17x - 20 (which simplifies to 2x⁴ + x³ + 17x - 20). This tool does the bookkeeping - lining up like terms and keeping track of signs and exponents - so you can check your work instantly.

How polynomial arithmetic works

Internally, every polynomial is just a list of coefficients indexed by the power of x. For example 2x^2 + 3x - 5 is the list [2, 3, -5] (highest degree first). The three operations are simple rules on these lists:

The multiplication rule is a convolution: each coefficient of the product is the sum of every pair of input coefficients whose exponents add to that power. It is the same calculation you do by hand with the distributive property (or FOIL for two binomials), just organized so nothing is missed.

A worked example: addition

Add (2x² + 3x - 5) and (x² - x + 4). Group the like terms:

• x² terms: 2x² + 1x² = 3x²
• x terms: 3x + (-1x) = 2x
• constants: -5 + 4 = -1

Putting them in order gives 3x² + 2x - 1. The degree stays at 2 because the leading terms did not cancel.

A worked example: multiplication

Multiply (x + 2) by (x - 3). Distribute every term of the first across the second: x·x + x·(-3) + 2·x + 2·(-3) = x² - 3x + 2x - 6. Combining the two x terms gives x² - x - 6. Notice the result is degree 2 - the sum of the two degree-1 inputs - which is always the case for multiplication.

A bigger multiplication: trinomial times binomial

Two-term products are easy enough to do in your head, but the bookkeeping grows fast once either polynomial has three or more terms. Multiply (x² + 2x - 1) by (3x - 4). Every one of the three terms in the first factor must hit both terms in the second, giving six partial products before any cleanup:

• x²·3x = 3x³ and x²·(-4) = -4x²
• 2x·3x = 6x² and 2x·(-4) = -8x
• -1·3x = -3x and -1·(-4) = +4

Now collect like terms: the x² column is -4x² + 6x² = 2x², and the x column is -8x - 3x = -11x. The standard-form answer is 3x³ + 2x² - 11x + 4, degree 3 (2 + 1, exactly as expected). This is precisely the kind of nine-or-fewer-product expansion where a sign or a like-term gets dropped by hand, so it is worth letting the calculator confirm your result.

FOIL and why it is just the distributive property

FOIL - First, Outer, Inner, Last - is a memory aid for multiplying two binomials, and nothing more. For (2x + 3)(x - 5) it tells you to take the First terms (2x·x = 2x²), the Outer terms (2x·-5 = -10x), the Inner terms (3·x = 3x), and the Last terms (3·-5 = -15), then combine: 2x² - 10x + 3x - 15 = 2x² - 7x - 15. The catch is that FOIL only works when both factors have exactly two terms. The moment you have a trinomial, FOIL has no "Outer" or "Inner" to point at, and you fall back on the general rule: multiply every term by every term. That general rule is what this calculator applies, so it never runs out of letters the way FOIL does.

Special products worth recognizing

A few products show up so often that memorizing their patterns saves time and helps you spot factoring opportunities later. You can verify each one with this calculator:

• Difference of squares: (a + b)(a - b) = a² - b². The middle terms cancel, so (x + 4)(x - 4) = x² - 16.
• Perfect-square trinomial: (a + b)² = a² + 2ab + b², so (x + 3)² = x² + 6x + 9. The middle term is twice the product of the two parts - forgetting it ("freshman's dream") is a classic error.
• Square of a difference: (a - b)² = a² - 2ab + b², so (2x - 1)² = 4x² - 4x + 1.
• Sum and difference of cubes: these come from products like (x + 2)(x² - 2x + 4) = x³ + 8, useful when you later need to factor a cubic.

Because the calculator squares a binomial by multiplying it by itself, you can confirm any of these expansions in a couple of seconds rather than trusting the pattern blindly.

Naming polynomials by terms and degree

Polynomials are classified two ways, and the vocabulary turns up constantly in algebra and pre-calculus. By the number of terms: a monomial has one term (5x²), a binomial has two (x + 3), and a trinomial has three (x² - x + 6); beyond that it is simply "a polynomial." By degree: degree 0 is a constant, degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, degree 4 is quartic, and degree 5 is quintic. These labels matter because the degree predicts a polynomial's shape and behavior - a quadratic graphs as a parabola, while higher-degree polynomials can wiggle up and down more times. When you need the roots of a quadratic specifically, the Quadratic Formula Calculator takes the standard-form output of this tool and solves it.

Where polynomial arithmetic shows up in the real world

Combining and expanding polynomials is not just classroom busywork - it underpins a lot of everyday quantitative work:

• Area and volume: if a rectangle is (x + 5) wide and (x + 2) tall, its area is the product (x + 5)(x + 2) = x² + 7x + 10. Adding a border or stacking shapes turns into adding polynomials.
• Finance and growth: compound-interest and annuity formulas expand into polynomials in the growth factor, and economists model cost, revenue, and profit as polynomial functions of quantity produced.
• Physics and motion: the height of a thrown object over time is a quadratic, and combining several motion equations means adding their polynomial pieces.
• Engineering and graphics: curves used in design and computer graphics (splines and Bezier curves) are built from polynomials that get added and multiplied to blend shapes smoothly.
• Statistics: fitting a trend line or a curve to data (polynomial regression) produces a polynomial you then evaluate and compare - related to the work the Average Calculator and Slope Calculator handle for simpler cases.

Why standard form matters

Writing the answer from the highest power down to the constant is not just a tidiness rule. Standard form lets you read the degree and leading coefficient at a glance, which together tell you the polynomial's end behavior - whether the graph rises or falls on the far left and right. It also makes two answers directly comparable: lined up by power, you can see instantly whether they are equal, and you can add or subtract them column by column without re-sorting. Most textbooks, graders, and software expect standard form, so the calculator always returns its result that way, even if you typed the terms out of order or with gaps. If you need to raise a single quantity to a power while you work, the Exponent Calculator handles that piece.

How to use this calculator

1. Enter the first polynomial using ^ for exponents, e.g. 3x^3 - 2x + 7. Spaces are optional.
2. Pick the operation: add, subtract, or multiply.
3. Enter the second polynomial in the same format.
4. Press Calculate. The big result at the top is the answer in standard form; below it you get the degree, a side-by-side of both inputs, and a term-by-term table.

You can leave off a coefficient of 1 (type x rather than 1x), use negative and decimal coefficients, and write the constant term as a plain number.

Who this is for

• Algebra 1 and 2 students checking homework on combining like terms and multiplying binomials.
• Pre-calculus and calculus students simplifying expressions before differentiating or integrating.
• Teachers and tutors who want a fast way to generate or verify answers in standard form.
• Anyone who needs to expand a product like (x + 2)(x - 3) without slips in sign or exponent.

Key terms explained

• Term: a single piece of a polynomial, such as 3x², made of a coefficient and a power of x.
• Coefficient: the number multiplying the variable - the 3 in 3x².
• Degree: the highest exponent with a non-zero coefficient.
• Like terms: terms with the same power of x, which can be combined by adding their coefficients.
• Standard form: terms written from highest to lowest exponent.
• Leading coefficient: the coefficient of the highest-degree term.

Three quick scenarios

• Simplifying before a derivative: expand (2x - 1)(x + 4) = 2x² + 7x - 4, then differentiate the simpler form.
• Combining areas: if two regions have areas x² + 2x and 3x + 5, their total area is the sum x² + 5x + 5.
• Checking a factoring answer: multiply your factors back together; if (x + 2)(x - 3) gives x² - x - 6, the factoring matches the original.

What affects the result

• Signs: a single dropped negative changes the whole answer - subtraction in particular flips every term of the second polynomial.
• Matching exponents: only terms with the same power combine; x² and x are not like terms.
• The operation: addition and subtraction keep (or lower) the degree, while multiplication adds the degrees together.
• Cancellation: opposite terms can vanish, sometimes dropping the degree of the answer.

Tips for working by hand

• Always rewrite subtraction as adding the opposite before combining terms.
• Stack like terms in columns so nothing is skipped, especially with missing powers (treat them as 0).
• After multiplying, double-check the degree equals the sum of the two input degrees.
• Finish by writing the answer in standard form, highest power first.

Limitations and assumptions

• It handles a single variable (x). Expressions like xy or x + y are not supported.
• Exponents must be whole numbers of 0 or greater - polynomials do not have fractional or negative powers.
• It performs add, subtract and multiply only - not division, factoring or solving.
• Decimal coefficients are kept, with tiny rounding applied only to remove floating-point noise.

How it compares to related calculators

This page handles polynomial arithmetic. For other algebra tasks, a sister tool fits better:

• To solve a degree-2 equation, use the Quadratic Formula Calculator.
• To raise a number to a power, use the Exponent Calculator.
• For trig, logs and general evaluation, use the Scientific Calculator.
• To find the slope and equation of a line, use the Slope Calculator.
• For percentages and the mean of a data set, see the Percentage Calculator and Average Calculator.