Polynomial Calculator
Enter one polynomial — or two — and get the sum, difference, product, quotient, full factorization, and every root in a single pass
Leave the second box empty to only factor and find roots of the first polynomial.
One Polynomial In, Every Operation Out
This polynomial calculator doesn’t make you pick a mode. Enter your expression once and get the sum, difference, product, division, factored form, and roots together.
Your Polynomial Breakdown
A Full Worked Example
Here’s exactly what the calculator does with 2x⁴ − 3x³ − 15x² + 32x − 12 divided by x² − 4x − 12 — every operation, one input.
Combine Like Terms
Adding the two polynomials lines up matching powers of x and adds coefficients term by term, giving 2x⁴ − 3x³ − 14x² + 28x − 24.
Distribute Every Term
Multiplying means every term of the first polynomial times every term of the second — 15 products in this case — then simplifying down to 2x⁶ − 11x⁵ − 27x⁴ + 128x³ + 40x² − 336x + 144.
Long-Divide the Leading Terms
Dividing the leading terms repeatedly gives a quotient of 2x² + 5x + 29 with a remainder of 208x + 336 left over the divisor.
Test Rational Root Candidates
Using the rational root theorem on the constant and leading coefficients narrows the search to a short list of p/q candidates — testing them by synthetic division finds x = 2 works first.
Repeat on the Reduced Polynomial
Dividing out the factor found in Step 4 leaves a lower-degree polynomial — the same root-testing process repeats until everything is fully factored.
Report Roots With Multiplicity
The final factorization (x−2)²(x+3)(2x−1) shows x = 2 is a double root — it touches the x-axis without crossing it, unlike the other two roots.
Why Trial-and-Error Factoring Falls Apart Past a Quadratic
Most students learn to factor quadratics by guessing two numbers that multiply and add correctly. That trick stops working the moment the degree climbs past 2 — and that’s exactly where this calculator’s method takes over.
| Degree | Guess-and-Check | Rational Root Theorem |
|---|---|---|
| Quadratic (degree 2) | Usually fine by hand | Confirms the guess instantly |
| Cubic (degree 3) | Guessing rarely finds the root | Narrows to a short p/q list |
| Quartic and higher | Practically impossible by guessing | Same method, repeated reduction |
| Repeated roots | Easy to miss a double root | Multiplicity tracked automatically |
Every Operation, Built On the Same Engine
No separate tools to juggle — one polynomial input drives all six results below.
Sum & Difference
Like terms — matching variable, matching exponent — combine by adding or subtracting just their coefficients. Unlike terms stay separate forever.
Product
Every term of the first expression distributes across every term of the second — a degree-4 by degree-2 multiplication produces 15 individual products before simplifying.
Quotient & Remainder
Long division repeatedly divides leading terms, multiplies back, and subtracts — stopping once what’s left has a lower degree than the divisor.
Rational Root Candidates
Factors of the constant term over factors of the leading coefficient generate every possible rational root — tested one by one with synthetic division.
Full Factorization
Each confirmed root divides cleanly out, reducing the degree by one — repeated until the polynomial is fully broken into linear and quadratic factors.
Roots With Multiplicity
A root appearing twice in the factorization, like (x−2)², means the graph touches the x-axis there instead of crossing — relevant in calculus when analyzing turning points.
Who Reaches for the Rational Root Theorem
Algebra 2 Students
First encounter with cubic and quartic factoring, where the “guess two numbers” trick from Algebra 1 stops being enough — this is where p/q candidate lists get introduced.
Precalculus & College Students
Verify factorizations and root multiplicities before graphing polynomial functions, where multiplicity determines whether a curve crosses or just touches the x-axis.
Teachers & Tutors
Generate a fresh worked example with a full p/q candidate list in seconds, instead of building one by hand for every lesson.
Frequently Asked Questions
What people actually ask when factoring and dividing polynomials.
Why does this calculator need a second polynomial box?
The second box is only needed for addition, subtraction, multiplication, and division, since those operations combine two expressions. Factoring and root-finding work on a single polynomial, so the second box can stay empty for those.
What is the rational root theorem and why does it matter here?
The rational root theorem says any rational root of a polynomial with integer coefficients must equal a factor of the constant term divided by a factor of the leading coefficient. Instead of guessing randomly, this generates a finite, short list of p/q candidates to test by synthetic division — the only practical way to factor a cubic or higher-degree polynomial by hand.
What does it mean when a root has multiplicity 2?
Multiplicity 2 means the same factor appears twice in the polynomial’s factorization, like (x − 2)². Graphically, the curve touches the x-axis at that point and bounces back, rather than crossing through it the way a multiplicity-1 root does.
How is dividing polynomials different from dividing two numbers?
The process mirrors long division: divide the leading terms, multiply that result by the entire divisor, subtract, and bring down the next term. Unlike number division, polynomial division can leave a remainder that is itself an expression, not just a number — written as a fraction over the original divisor.
Can a polynomial have no rational roots at all?
Yes. The rational root theorem only finds rational candidates — many polynomials have irrational or complex roots that no amount of p/q testing will reveal. When every rational candidate fails, the remaining factor is typically solved with the quadratic formula or left in its current form.
Does multiplying two polynomials always increase the degree?
Yes, multiplying a degree-m polynomial by a degree-n polynomial always produces a degree (m + n) result, since the highest-power terms multiply together and can’t be canceled by anything else in the product.
Why does the remainder matter when dividing polynomials?
A remainder of zero means the divisor is a clean factor of the dividend — that’s exactly how the rational root theorem confirms each candidate root. A nonzero remainder simply means the division doesn’t come out even, which is a normal and expected result.
Does this work for polynomials with more than one variable?
Addition, subtraction, and multiplication work fine for multivariate polynomials. Long division, the rational root theorem, and root-finding are designed specifically for single-variable polynomials and won’t apply the same way to expressions with two or more variables. For broader multivariable work, see our algebra solver.
What Makes This Polynomial Calculator Different
Most polynomial tools split addition, multiplication, division, and factoring into separate calculators, forcing you to copy your expression from one page to another. This one runs every operation against the same input: combine two polynomials, divide them, and fully factor and find the roots of the result — all from a single entry. It’s one of several free math solver tools built around showing the complete reasoning, not just a final number.
The factoring engine specifically applies the rational root theorem, the method that actually scales past quadratics — where the “find two numbers” shortcut taught in early algebra stops working.
Combining Polynomials: Addition and Subtraction
Addition and subtraction only touch like terms — terms sharing the same variable raised to the same exponent. Adding (2x⁴ − 3x³ − 15x² + 32x − 12) and (x² − 4x − 12) means lining up the x² terms (−15 + 1 = −14), the x terms (32 − 4 = 28), and the constants (−12 − 12 = −24), leaving the x⁴ and x³ terms untouched since the second polynomial has no matching terms. The result: 2x⁴ − 3x³ − 14x² + 28x − 24.
Subtraction works the same way, except every term in the second polynomial flips sign first — a step that’s easy to apply only to the first term by mistake.
Multiplying Polynomials of Any Size
For two binomials, the FOIL method — First, Outer, Inner, Last — covers all four products needed. But multiplying a degree-4 polynomial by a degree-2 polynomial means distributing all 5 terms across all 3, producing 15 individual products before anything simplifies. Each term of the first polynomial multiplies every term of the second, and the results combine afterward by exponent, the same way addition works.
One shortcut worth knowing: multiplying a degree-m polynomial by a degree-n polynomial always produces a degree (m + n) result, since the two leading terms multiply into a new highest-degree term that nothing else in the product can cancel.
Long Division: Quotient and Remainder
Polynomial long division mirrors numerical long division: divide the leading term of the dividend by the leading term of the divisor, multiply that result by the whole divisor, subtract from the dividend, and bring down the next term. Repeat until what remains has a lower degree than the divisor.
Dividing 2x⁴ − 3x³ − 15x² + 32x − 12 by x² − 4x − 12 produces a quotient of 2x² + 5x + 29 with a remainder of 208x + 336 — written as a fraction over the original divisor in the complete answer.
Factoring Past the Quadratic: the Rational Root Theorem
Once a polynomial’s degree climbs to 3 or higher, “guess two numbers that multiply and add correctly” no longer works — there are simply too many combinations to try by intuition. The rational root theorem narrows the search instead: list every factor of the constant term (call them p), every factor of the leading coefficient (call them q), and every possible value of p/q becomes a candidate root.
For 2x⁴ − 3x³ − 15x² + 32x − 12, the constant term −12 has factors ±1, ±2, ±3, ±4, ±6, ±12, and the leading coefficient 2 has factors ±1, ±2. That generates candidates like ±1, ±2, ±3, ±½, ±3⁄2, and more — each one tested by synthetic division until the remainder hits zero.
Repeating the Process
Once one root is confirmed, dividing it out reduces the polynomial’s degree by one. The same candidate-testing process repeats on the smaller polynomial until everything is fully factored — for this example, down to (x − 2)²(x + 3)(2x − 1).
Reading Multiplicity in the Final Factorization
When the same factor shows up more than once, like (x − 2)² in the example above, that root has multiplicity 2. Graphically, a multiplicity-1 root crosses straight through the x-axis, while a multiplicity-2 root touches the axis and bounces back without crossing — a detail that matters when graphing the function or analyzing its turning points in calculus.
Mistakes That Show Up Most Often
- Combining unlike terms: x² and x can never merge into one term, no matter how tempting it looks.
- Flipping the sign on only the first term when subtracting: every term of the second polynomial needs the sign change, not just the leading one.
- Guessing factors past degree 2: the rational root theorem exists precisely because intuition runs out once a polynomial passes quadratic.
- Reporting a double root as if it were two separate values: multiplicity 2 is one root with extra graphical significance, not two distinct roots.