What is a polynomial?

A polynomial is an expression made of terms, where each term is a coefficient times a non-negative integer power of a variable. For example, 2x^3 minus 5x plus 7 is a polynomial of degree 3. The degree is the highest power of x that appears; the leading coefficient is the number in front of that highest power.

How do I multiply two polynomials?

Multiply every term of the first polynomial by every term of the second, then combine like terms. For two binomials this is the FOIL pattern (First, Outer, Inner, Last). For larger polynomials, distribute each term of the first across every term of the second and add up products of matching degrees.

What is the difference between long division and synthetic division?

Long division of polynomials works for any divisor and follows the same divide-multiply-subtract-bring-down pattern as long division of integers. Synthetic division is a faster shortcut that only works when the divisor is a linear monic factor of the form (x minus c). For divisors of degree 2 or higher, or with leading coefficient other than 1, you must use long division.

How do I factor a polynomial?

Try methods in this order: first pull out the greatest common factor, then attempt grouping if there are four or more terms, then check for special forms like difference of squares or sum and difference of cubes, then use the rational root theorem to find linear factors of higher-degree polynomials. For any root found, divide it out and continue factoring the quotient.

How do I find all the zeros of a polynomial?

By the fundamental theorem of algebra, a polynomial of degree n has exactly n complex zeros, counted with multiplicity. Find rational candidates using the rational root theorem (p divides the constant, q divides the leading coefficient, candidates are plus or minus p over q), test each, and for every root found divide it out and continue on the smaller quotient. The remaining quadratic or smaller factor can be solved with the quadratic formula or further factoring.

Polynomial Calculator: All Operations

Key Terms

Polynomial — an expression of the form

a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

where each

a_i

is a number and

n

is a non-negative integer.

Degree — the highest power of

x

that appears with a non-zero coefficient. The degree determines a polynomial's end behavior and the maximum number of real roots.

Coefficient — the numerical factor in front of a power of

x

. The leading coefficient is the coefficient of the highest-degree term.

Standard form — the polynomial written with terms ordered by descending degree, no like terms left uncombined, and no explicit zero terms.

Root (or zero) — a value

x = c

where

P(c) = 0

. The roots of

P

are exactly the values where the graph of

y = P(x)

crosses the x-axis (counting multiplicity for tangent crossings).

Factor — a polynomial

Q(x)

such that

P(x) = Q(x) \cdot R(x)

for some polynomial

R(x)

. Linear factors correspond to roots:

(x - c)

is a factor exactly when

c

is a root.

Long division — the general algorithm for dividing one polynomial by another. Works for any divisor.

Synthetic division — a shortcut for dividing by a linear monic divisor

(x - c)

. Faster but more limited in scope.

Quotient and remainder — when

P_1

is divided by

P_2

, the result

Q

and

R

satisfy

P_1 = P_2 \cdot Q + R

with

\deg R < \deg P_2

.

Getting Started

The calculator opens on the Add operation with two starter polynomials loaded. To work with a different operation, click any tab in the operation strip at the top: *Add, Subtract, Multiply, Long division, Synthetic, Factor, Find zeros, Simplify*.

The layout has three main areas:

• Operation strip at the top selects which operation runs. The page title and tooltip update to match the selected operation.

• Input card on the left holds your polynomial(s). For single-input operations (Factor, Find zeros, Simplify) there is one input box. For division (Long, Synthetic) there are two labeled boxes — Dividend and Divisor. For Add, Subtract, and Multiply you can have two or more inputs and add more with the + Add another polynomial button.

• Solution & explanation card on the right shows the step-by-step work, the method description, common pitfalls, and related tools.

To switch input modes, use the Keypad / Slots toggle above the polynomial list. The two modes are interchangeable for the same polynomial — pick whichever feels faster for the polynomial you're entering.

The URL supports an 'op' query parameter: append '?op=3' to land directly on the Multiply tab, '?op=4' for Long division, and so on. The numbers follow the order of the operation strip.

The Eight Operations

The calculator covers the full set of standard polynomial operations:

• Add — combine two or more polynomials, summing coefficients of matching degrees.

• Subtract — distribute the minus sign across every term of the subtrahend, then add.

• Multiply — expand the product of two or more polynomials with full distribution and like-term collection.

• Long division — general division algorithm, returns quotient and remainder for any divisor.

• Synthetic division — fast shortcut when the divisor is linear and monic,

(x - c)

.

• Factor — break the polynomial into irreducible factors using GCF, grouping, special forms, and rational roots in sequence.

• Find zeros — list every real and complex root with its multiplicity.

• Simplify — combine like terms, drop zero coefficients, and rewrite in standard form.

Each operation has its own demo inputs that load when you switch to it, so you can see a worked example immediately. The Clear button restores those defaults.

For each operation the calculator reports the primary result, optional extras (e.g., the remainder in division), and meta info like degree, leading coefficient, and constant term.

Input Modes — Keypad and Slots

The calculator offers two ways to enter a polynomial. Both produce the same internal representation, so you can switch freely.

Keypad mode is a calculator-style interface:

• A digit grid for typing coefficients and a row of operators (+, −, ×, ÷, ^, parentheses).

• A power dial on the left for inserting

x

at any chosen exponent. The dial has − and + step buttons and a row of preset chips for

x

,

x^2, \ldots, x^6

. For higher exponents, use the Custom power input.

• The Tap to insert button drops

x

at the current dial power directly into the expression. The display shows your polynomial in formatted math notation as you build it.

• A live validity check — the Save button greys out if the expression doesn't parse.

Slots mode is faster when you know the polynomial as a coefficient list:

• One numeric slot per power of

x

, from highest degree down to the constant. Each slot is labeled with its monomial like

x^3

or

x^2

.

• Degree + / − controls grow or shrink the list. Negative coefficients turn the slot red; zero coefficients turn grey.

• A live Reads as preview shows the polynomial built from the current slot values in math notation.

Click any polynomial box in the input list to open the editor in your current mode. Press Save to commit, Cancel to discard.

Reading the Step-by-Step

Every operation produces a sequence of numbered steps in the Step-by-step section. The number of steps is shown in the badge next to the section title.

Each step has three parts:

• Title — a short description of what the step accomplishes ("Line up by degree", "Apply FOIL", "Bring down the next term").

• Math — the formal manipulation in proper notation, shown in a highlighted code-style block.

• Note — a plain-language explanation of why the step works or what to watch for. Optional; not every step has one.

When the step list is long, a fading gradient and a bouncing chevron arrow appear at the bottom of the section to indicate scrollable content. Click the chevron or scroll directly to see more steps.

Below the step list, three more sections give context:

• About this method — a short prose description of the algorithm and the relevant formula.

• Common pitfalls — a bullet list of the most frequent mistakes for this operation.

• Related tools — chips linking to related calculators and theory pages.

The result card on the left shows the primary answer along with Copy LaTeX, Share, and (for graphable results) a Graph button.

Adding and Subtracting Polynomials

Polynomial addition and subtraction are term-by-term. The rule is simple: combine the coefficients of like terms (terms with the same power of

x

), and keep all other terms unchanged.

Addition example. For

P_1(x) = 2x^2 + 3x - 1

and

P_2(x) = x^2 - 5x + 4

:

P_1 + P_2 = (2 + 1) x^2 + (3 - 5) x + (-1 + 4) = 3x^2 - 2x + 3

The degree of the sum is at most the maximum of the inputs' degrees:

\deg(P_1 + P_2) \le \max(\deg P_1, \deg P_2)

It can be strictly less if the leading terms cancel.

Subtraction example. For

P_1(x) = 3x^2 + 2x - 5

and

P_2(x) = x^2 - 4x + 1

:

P_1 - P_2 = (3 - 1) x^2 + (2 - (-4)) x + (-5 - 1) = 2x^2 + 6x - 6

The key step in subtraction is distributing the minus sign across every term of

P_2

. The single most common error is negating only the leading term and leaving the rest untouched. The calculator shows the distribution explicitly in step 1 of every subtraction.

The Add and Subtract operations both support more than two inputs — use + Add another polynomial to extend the chain.

Multiplying Polynomials

Polynomial multiplication uses the distributive property: every term of the first polynomial multiplies every term of the second, then like terms collect.

For two binomials, the FOIL mnemonic captures the four products: First, Outer, Inner, Last. For larger polynomials the principle is the same but extended:

(a_0 + a_1 x + \ldots)(b_0 + b_1 x + \ldots) = \sum_{i, j} a_i b_j \, x^{i+j}

The degree of the product is the sum of the input degrees:

\deg(P_1 \cdot P_2) = \deg P_1 + \deg P_2

(Assuming neither leading coefficient is zero.)

For three or more polynomials, the calculator multiplies pairwise from the left. Multiplication is associative, so any grouping yields the same final answer:

P_1 \cdot P_2 \cdot P_3 = (P_1 \cdot P_2) \cdot P_3 = P_1 \cdot (P_2 \cdot P_3)

Left-to-right grouping is usually fastest because each intermediate product stays as small as possible.

For a visual treatment of pairwise multiplication, see the polynomial multiplication visualizer, which shows every

a_i b_j

product as a colored cell in a grid, then collects like-term diagonals into the final answer.

Polynomial Division — Long and Synthetic

Polynomial division produces a quotient

Q(x)

and a remainder

R(x)

satisfying:

P_1(x) = P_2(x) \cdot Q(x) + R(x), \qquad \deg R < \deg P_2

The calculator implements two division algorithms.

Long division works for any divisor. The procedure mirrors long division of integers:

• Divide leading terms to get the next quotient term.

• Multiply the divisor by that term, subtract from the current dividend.

• Bring down the next term.

• Repeat until the remainder degree falls below the divisor's.

Synthetic division is a fast shortcut for dividing by a linear monic divisor

(x - c)

. It uses only the coefficients of the dividend and the value of

c

, skipping the explicit

x

terms.

• Write the dividend's coefficients (including placeholder zeros for missing degrees).

• Bring down the first coefficient.

• Multiply by

c

, add to the next coefficient. Repeat.

• The last value is the remainder; the rest form the quotient's coefficients.

When synthetic division does not work: any time the divisor is not of the form

(x - c)

. A divisor like

x^2 - 1

,

2x - 3

, or

(x - 1)(x + 1)

requires long division. The calculator's Synthetic tab assumes the divisor is monic and linear; for anything else, switch to Long division.

A useful corollary — the remainder theorem: when dividing

P(x)

by

(x - c)

, the remainder equals

P(c)

. This makes synthetic division a fast way to evaluate a polynomial at a point.

Factoring and Finding Zeros

Factoring and finding zeros are two sides of the same coin. Every linear factor

(x - c)

corresponds to a zero

x = c

; every zero

x = c

gives a factor

(x - c)

. The calculator runs the same underlying algorithm for both and reports the result in different formats.

Factoring strategies (the order the calculator tries them):

• Greatest common factor (GCF) — pull out the largest monomial that divides every term.

• Grouping — arrange terms in pairs and factor each pair; if the pairs share a binomial, factor it out.

• Special forms — difference of squares

a^2 - b^2 = (a-b)(a+b)

, sum/difference of cubes, perfect square trinomials.

• Rational root theorem — for higher-degree polynomials, candidate rational roots are

\pm p/q

where

p

divides the constant and

q

divides the leading coefficient. Test each candidate; for any root found, divide it out and continue on the lower-degree quotient.

Fundamental theorem of algebra: a polynomial of degree

n

has exactly

n

complex roots, counted with multiplicity. Real polynomials of odd degree always have at least one real root.

Zeros output format: the calculator lists every root with its multiplicity, separating real and complex roots in the meta info. A double root like the "1" in

(x - 1)^2 (x + 2)

counts twice in the multiplicity total but is listed once with its multiplicity called out.

The Common pitfalls card for these operations flags issues like stopping too early, forgetting complex roots, and counting multiplicities incorrectly.

Simplifying Polynomials

Simplifying a polynomial means rewriting it in standard form: terms in descending degree order, like terms combined, zero terms removed.

For example:

3x + 2x^2 - 5 + x - 2x^2 + 4

Group by degree:

•

x^2

:

2 - 2 = 0

(cancels out completely)

•

x

:

3 + 1 = 4

• constant:

-5 + 4 = -1

The simplified form is

4x - 1

. The degree dropped from 2 to 1 because the

x^2

terms cancelled.

The general standard form for a degree-

n

polynomial is:

a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0

with

a_n \ne 0

. Coefficients of cancelled terms are not written, but if a middle term has coefficient zero you typically omit it entirely rather than writing

0 \cdot x^k

.

When to use Simplify vs. other operations: Simplify takes a single messy polynomial and tidies it up. If you have two or more polynomials to combine, use Add (or Subtract or Multiply) instead — those operations end in a simplified result anyway.

Related Concepts and Tools

Polynomial multiplication visualizer — an interactive grid showing every pairwise product as a colored cell, with like-term diagonals collected into the final answer. Useful for building intuition before trusting the calculator's step-by-step.

Binomial coefficient visualizer — three views of

(a + b)^n

: a decision tree, an animated product expansion, and Pascal's triangle as a path counter. The right place to internalize where the coefficients in

(a + b)^n

come from.

Completing the square visualizer — for quadratic polynomials, the geometric reading of the algebraic technique that converts

ax^2 + bx + c

to vertex form.

Rational root theorem — theoretical underpinning for the Factor and Find zeros operations on higher-degree polynomials.

Remainder theorem — the corollary of polynomial division that says

P(c)

equals the remainder when dividing by

(x - c)

. Provides the link between division and evaluation.

Fundamental theorem of algebra — the existence statement behind "degree

n

polynomials have

n

complex roots."

Calculator extensions on calculatematrix.com — for polynomials in multiple variables, polynomial GCD, partial fraction decomposition, and other operations not in this calculator, see the dedicated polynomial calculators page on the sister site.