Pick an operation in the strip at the top: Add, Subtract, Multiply, Long division, Synthetic, Factor, Find zeros or Simplify; each loads its example polynomials, result and steps. Learn more about getting started
Read the title and the ? tooltip to see what the operation does; the Result card shows the primary answer with its degree, leading coefficient and constant. Learn more about the eight operations
Click a polynomial box to open the editor in the current mode, Keypad or Slots, and press Save or Cancel; the toggle above the list switches the mode. Learn more about the input modes
Read the Solution & explanation card on the right: the numbered steps, the note on the operation, About this method, and Common pitfalls. Learn more about the step-by-step
Polynomial — an expression of the form anxn+an−1xn−1+⋯+a1x+a0 where each ai 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)⋅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 P1 is divided by P2, the result Q and R satisfy P1=P2⋅Q+R with degR<degP2. The words are defined at length in Vocabulary and Degree of the polynomials lesson.
Getting Started
DemoThree operations from the strip
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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 and the common pitfalls.
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 Share button on the result card builds a link of this kind with your polynomials added, so whoever opens it sees the same inputs and the same result. The strip holds eight buttons: Add, Subtract, Multiply, Long division, Synthetic, Factor, Find zeros and Simplify; clicking one loads that operation's example polynomials, its result and its steps, so the Subtract button shows 3x2+2x−5 and x2−4x+1 with the difference 2x2+6x−6, and Long division shows a Dividend and a Divisor with the quotient x2−x−1 and the remainder 3. What each operation does is explained in The Eight Operations and in the theory sections below, starting with Adding and Subtracting Polynomials.
The Eight Operations
DemoMultiply, Factor, Simplify
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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 — pull out the greatest common factor, then use the rational root theorem to split off every linear factor; what is left with no rational root stays as one factor.
• Find zeros — list every real and complex root with its multiplicity: rational roots exactly, the rest from the quadratic formula or a numeric method.
• 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. Multiply opens with three polynomials, (x+2)(x−3)(2x+1)=2x3−x2−13x−6, and is one of the operations with a Graph button; Factor opens with one box, x3−2x2−x+2=(x+1)(x−1)(x−2); Simplify opens with 3x+2x2−5+x−2x2+4=4x−1. Everything on the result card is computed from the boxes: edit a polynomial, press the operation button, and the result and the steps are worked out for your polynomials, not for the example. Until you press the button the card still shows the previous answer, and a short note under the buttons says so. Brackets, products like (x+1)(x−2), powers and division by a number are all accepted; an input that cannot be read gives a message in place of the result. The mathematics of every operation is explained in its own section, from Adding Polynomials to Simplifying Polynomials.
Input Modes — Keypad and Slots
DemoThe Keypad editor, then the Slots editor
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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, x2,…,x6. 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 x3 or x2.
• 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. The editor header names the box and the mode, for example "Editing Polynomial 1 · Keypad"; Cancel closes it with the polynomial unchanged, and the Keypad / Slots toggle decides which editor the next click opens. The notation the slots use, one coefficient per power, is explained in Simplifying Polynomials.
Adding, Removing and Resetting Polynomials
DemoTwo boxes added, then Clear
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For Add, Subtract and Multiply the input card works on a list of polynomials, and the + Add another polynomial button under the list appends a new box holding x+1, labelled P3, P4 and so on, with the operation sign between the boxes. Each box beyond the second has a ✕ button on its right that removes it; with only two boxes left the button is hidden, since every one of these operations needs two inputs. Long division and Synthetic always have exactly two boxes, Dividend and Divisor, and Factor, Find zeros and Simplify have one, so the add button is not shown for them.
The two buttons under the list, the operation button labelled with the operation's name and Clear, finish the card. Clear puts back the operation's example polynomials, two for Add, three for Multiply, and removes any boxes you added. The operation button works out the result and the steps for the polynomials in the boxes; if you have changed a box and not yet pressed it, a note under the buttons reminds you. Clear also resets the result to the example. Where the two example polynomials of Add come from and how they are summed is explained in Adding Polynomials.
Reading the Step-by-Step
DemoThe steps for Multiply, Synthetic and Find zeros
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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, two 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.
The result card on the left shows the primary answer along with Copy LaTeX (copies the answer as LaTeX), Share (copies a link that reopens the same operation with your polynomials) and, for the graphable operations, a Graph button that draws the polynomial with its real zeros marked. Switching to Multiply changes the badge to that operation's step count and the About this method text to the pairwise expansion rule; Synthetic adds the Remainder3 as an extra under the quotient; Find zeros lists x=−1, x=1, x=2 with the meta info "Real roots: 3". The steps are written for the polynomials in the boxes, so a different input gives different steps and a different step count. The steps of each operation are explained in the theory sections, for example Polynomial Long Division.
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 P1(x)=2x2+3x−1 and P2(x)=x2−5x+4:
P1+P2=(2+1)x2+(3−5)x+(−1+4)=3x2−2x+3
The degree of the sum is at most the maximum of the inputs' degrees:
deg(P1+P2)≤max(degP1,degP2)
It can be strictly less if the leading terms cancel.
Subtraction example. For P1(x)=3x2+2x−5 and P2(x)=x2−4x+1:
P1−P2=(3−1)x2+(2−(−4))x+(−5−1)=2x2+6x−6
The key step in subtraction is distributing the minus sign across every term of P2. 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.
To add polynomials, line up the terms with the same power of x and add their coefficients; the powers themselves do not change. The example loaded by the Add operation is P1=2x2+3x−1 and P2=x2−5x+4: the x2 coefficients give 2+1=3, the x coefficients 3+(−5)=−2, the constants −1+4=3, so P1+P2=3x2−2x+3, of degree 2 with leading coefficient 3.
The degree of a sum is at most the larger of the two degrees, and it is smaller when the leading terms cancel, as in (x2+x)+(−x2+1)=x+1. The common mistake is to add unlike terms, treating x2 and x as if they combined; a missing power is a zero coefficient and stays zero.
Subtraction is addition of the opposite: P1−P2=P1+(−P2), so every sign in the second polynomial flips before the like terms are combined. The Subtract example is P1=3x2+2x−5 and P2=x2−4x+1: flipping gives −x2+4x−1, and adding gives (3−1)x2+(2+4)x+(−5−1)=2x2+6x−6.
The common mistake, and the one the first step of the calculator makes visible, is to negate only the first term of P2 and leave the others as they are, which here would give 2x2−2x−6 instead. Subtraction is not commutative: P2−P1=−2x2−6x+6, the negative of the answer above.
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:
(a0+a1x+…)(b0+b1x+…)=i,j∑aibjxi+j
The degree of the product is the sum of the input degrees:
deg(P1⋅P2)=degP1+degP2
(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:
P1⋅P2⋅P3=(P1⋅P2)⋅P3=P1⋅(P2⋅P3)
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 aibj product as a colored cell in a grid, then collects like-term diagonals into the final answer. The lesson works the method through in Multiplying polynomials, the FOIL method and Special products; the degree rule is in Degree and operations.
Polynomial Division — Long and Synthetic
Polynomial division produces a quotientQ(x) and a remainderR(x) satisfying:
P1(x)=P2(x)⋅Q(x)+R(x),degR<degP2
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 x2−1, 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. Each algorithm has its own section with the example x3−2x2+4 divided by x−1: Polynomial Long Division and Synthetic Division. The lesson covers them in Polynomial long division and Synthetic division, and the corollary in The remainder theorem.
Polynomial Long Division
Long division divides one polynomial by another of lower or equal degree and returns a quotient Q and a remainder R with P1=P2⋅Q+R and degR<degP2. Each round divides the leading terms, multiplies the divisor by that term, subtracts, and brings down the next term. The example is x3−2x2+4 divided by x−1: x3÷x=x2, subtracting x2(x−1) leaves −x2+0x; then −x, leaving −x+4; then −1, leaving the remainder 3. So Q=x2−x−1, R=3, and the identity reads x−1x3−2x2+4=x2−x−1+x−13.
The missing x term of the dividend is written as 0x so that the columns line up; forgetting the placeholder is the common mistake. Long division works for any divisor, including x2−1 or 2x−3, which synthetic division cannot handle.
Synthetic division is the shortcut for dividing by a monic linear divisor x−c. Write the coefficients of the dividend with placeholder zeros, bring down the first, then repeat: multiply by c and add to the next coefficient. For x3−2x2+4 and x−1, so c=1: the row 1,−2,0,4 gives 1; 1⋅1+(−2)=−1; −1⋅1+0=−1; −1⋅1+4=3. The first three numbers are the quotient x2−x−1 and the last is the remainder 3, the same answer as long division in a third of the writing.
The remainder equals P(c): here P(1)=1−2+4=3, which is the remainder theorem and the reason synthetic division is also a fast way to evaluate a polynomial. The common mistake is to use the sign of the divisor as written, −1 instead of c=1 for x−1, which gives a wrong row; the Synthetic tab of the calculator expects a monic divisor and falls back to long division otherwise.
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 a2−b2=(a−b)(a+b), sum/difference of cubes, perfect square trinomials.
• Rational root theorem — for higher-degree polynomials, candidate rational roots are ±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.
Factoring writes a polynomial as a product of polynomials of lower degree, down to factors that cannot be split further over the real numbers. The calculator first pulls out the greatest common factor, then tests the rational root candidates and divides each root out; a leftover quadratic is split when its discriminant is a perfect square, and any other leftover stays as one factor. Grouping and special forms are done by hand, as in the example x3−2x2−x+2 factors by grouping: x2(x−2)−(x−2)=(x2−1)(x−2), and the difference of squares finishes it, (x−1)(x+1)(x−2). Multiplying back, (x−1)(x+1)=x2−1 and (x2−1)(x−2)=x3−2x2−x+2, checks the answer.
Each linear factor x−c names a zero c of the polynomial, so the factored form is also the list of zeros, which is why the Factor and Find zeros operations share one algorithm. The common mistake is to stop at (x2−1)(x−2) with a factor that still splits.
A zero of P is a value c with P(c)=0. For a polynomial with integer coefficients the rational root theorem lists the candidates, ±p/q with p dividing the constant and q dividing the leading coefficient; for x3−2x2−x+2 the candidates are ±1 and ±2, and testing gives P(1)=0, P(−1)=0, P(2)=0. So the zeros are x=−1, x=1, x=2, three real roots and no complex ones, matching the factored form (x+1)(x−1)(x−2). A root found by testing is divided out by synthetic division and the search continues on the quotient.
The fundamental theorem of algebra fixes the count: a polynomial of degree n has exactly n complex roots counted with multiplicity, so a cubic with three real roots has no complex ones left. The common mistake is to forget the multiplicity, counting the double root of (x−1)2(x+2) once.
Simplifying a polynomial means rewriting it in standard form: terms in descending degree order, like terms combined, zero terms removed.
For example:
3x+2x2−5+x−2x2+4
Group by degree:
• x2: 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 x2 terms cancelled.
The general standard form for a degree-n polynomial is:
anxn+an−1xn−1+…+a1x+a0
with an=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⋅xk.
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. Standard form and the vocabulary around it are taught in Standard form and Vocabulary of the polynomials lesson.
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 ax2+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.
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. The theorems named here are collected in Key theorems and in the Master toolkit of polynomial theorems.