Polynomial Calculator

Q: What operations does the polynomial calculator support?

Expanding, adding, subtracting, multiplying, long division with quotient and remainder, evaluating at a value of x using Horner's method, and finding real roots.

Q: How do I divide polynomials?

Choose divide, enter the dividend as P and the divisor as Q. The calculator performs long division and shows the quotient, the remainder and the answer in the form quotient plus remainder over divisor.

Q: What is the remainder theorem?

When a polynomial P is divided by x minus c, the remainder equals P(c). So evaluating the polynomial at c is a quick way to find the remainder or test whether c is a root.

Q: Does the polynomial calculator list complex roots?

No. Roots mode lists real roots only and states when a polynomial has none. Rational roots are exact and irrational roots are shown as decimal approximations.

Q: What is Horner's method?

A way of evaluating a polynomial by nesting multiplications, such as writing ax cubed plus bx squared plus cx plus d as ((ax + b)x + c)x + d. It needs fewer operations and is less error-prone than computing each power separately.

Q: What is the degree of a polynomial?

The highest power of the variable with a nonzero coefficient. It limits the number of real roots and turning points the polynomial can have.

Q: Can I use fractions as coefficients?

Yes. Fractions and decimals are converted to exact fractions, and all results are kept exact.

About the Polynomial Calculator

Polynomials are the most common functions in algebra: lines, parabolas, cubics and everything built from whole-number powers of a variable. Working with them means a handful of operations that come up again and again, such as multiplying two brackets, dividing by a linear factor, plugging in a value or locating the zeros. Each is mechanical, but each is easy to get wrong when the polynomial has many terms.

This calculator performs seven operations on polynomials in one variable: expand, add, subtract, multiply, long division, evaluation at a point, and real roots. Coefficients are stored as exact fractions, so long division produces an exact quotient and remainder, and rational roots are reported exactly rather than as rounded decimals.

What is a polynomial?

A polynomial in $x$ is a sum of terms of the form $a x^n$ , where each coefficient $a$ is a number and each exponent $n$ is a whole number. $2x^3 - 3x^2 + 4x - 5$ is a polynomial; $\frac1x$ and $\sqrt x$ are not, because their exponents are negative or fractional.

The degree is the highest exponent with a nonzero coefficient, and the leading coefficient is the coefficient of that term. A polynomial of degree $n$ has at most $n$ real roots and its graph has at most $n - 1$ turning points, facts that help predict what an answer should look like before computing it.

How to use the Polynomial Calculator

Choose an operation: expand, add, subtract, multiply, divide, evaluate or roots.
Enter the first polynomial P, for example 2x^3 - 3x^2 + 4x - 5.
For add, subtract, multiply and divide, enter the second polynomial Q, such as x - 2.
For evaluate, enter the value of $x$ in the extra box.
Press Calculate and follow the steps; long division shows each leading-term division and subtraction.

Formulas

Division algorithm

P(x) = Q(x)\,S(x) + R(x),\quad \deg R < \deg Q

Remainder theorem

P(x) \div (x - c) \;\Rightarrow\; R = P(c)

Factor theorem

P(c) = 0 \iff (x - c) \text{ is a factor of } P(x)

Horner's method

a_3x^3 + a_2x^2 + a_1x + a_0 = ((a_3x + a_2)x + a_1)x + a_0

Worked example: Divide 2x³ − 3x² + 4x − 5 by x − 2

The example divides $2x^3 - 3x^2 + 4x - 5$ by $x - 2$ using long division. At each stage the engine divides the leading term of what remains by $x$ , multiplies the divisor by that term and subtracts.

Three rounds give the quotient $2x^2 + x + 6$ and the remainder 7, so the answer is written $2x^2 + x + 6 + \frac{7}{x - 2}$ . As the remainder theorem predicts, evaluating the original polynomial at $x = 2$ also gives 7.

The solution below is generated by the same engine that powers the calculator above.

Final answer

2x^{2} + x + 6 + \frac{7}{x - 2}

How we read your problem

Divide P by Q using polynomial long division.

Step-by-step solution

Write P in standard form
Expand and combine like terms: degree 3, leading coefficient 2, constant term -5.
$P(x) = 2x^{3} - 3x^{2} + 4x - 5$
Write Q in standard form
degree 1, leading coefficient 1, constant term -2
$Q(x) = x - 2$
Divide the leading terms: (2x^3) ÷ (x) = 2x^2
Multiply Q by 2x^2 and subtract. New remainder: x^2 + 4x - 5.
$2x^{2}\cdot(x - 2)$
Divide the leading terms: (x^2) ÷ (x) = x
Multiply Q by x and subtract. New remainder: 6x - 5.
$x\cdot(x - 2)$
Divide the leading terms: (6x) ÷ (x) = 6
Multiply Q by 6 and subtract. New remainder: 7.
$6\cdot(x - 2)$
Stop when the remainder's degree is less than the divisor's
$\text{quotient } 2x^{2} + x + 6,\quad \text{remainder } 7$

Formulas used

Division algorithm
$P(x) = Q(x)\,q(x) + r(x),\; \deg r < \deg Q$

Q does not divide P evenly: the remainder is 7.

Verification

Check: Q·quotient + remainder equals P.

Detailed explanation

Multiplying and expanding

To multiply two polynomials, every term of the first multiplies every term of the second, and like powers are then collected. For $(x + 1)(x^2 - x + 1)$ the six products collapse to $x^3 + 1$ , a sum of cubes. The calculator shows the full distribution before collecting so that no product is missed, which is the usual source of errors with longer brackets.

Adding and subtracting only combine coefficients of equal powers. With subtraction, the sign of every term of the second polynomial changes, which is why the calculator writes the negated polynomial explicitly before combining.

Long division and its uses

Polynomial long division works exactly like long division of numbers. It stops when the remainder has a smaller degree than the divisor. Division is used to factor a polynomial once one root is known, to simplify rational functions, and to find slant asymptotes: the quotient of $\frac{x^2 + 1}{x - 1}$ is $x + 1$ , so the graph approaches the line $y = x + 1$ .

Dividing by a linear factor $x - c$ can be shortened to synthetic division, which keeps only the coefficients. The remainder theorem then says the remainder equals $P(c)$ , a quick way to check your work.

Finding roots

In roots mode the calculator first tests rational candidates $\frac pq$ , where $p$ divides the constant term and $q$ divides the leading coefficient. Each root found is divided out, lowering the degree. A remaining quadratic is solved with the quadratic formula, and any irreducible remainder of higher degree is solved numerically, with the result marked as an approximation. For $x^3 - 6x^2 + 11x - 6$ the roots 1, 2 and 3 are all exact; for $x^3 - 2$ the single real root $\sqrt[3]{2} \approx 1.2599$ is shown as a decimal.

Only real roots are listed. A polynomial such as $x^2 + 2x + 5$ has a negative discriminant, so the calculator reports that it has no real roots.

Common mistakes

Leaving out a missing power in long division; write $x^3 + 0x^2 + 0x - 8$ so columns line up.
Subtracting only the first term during long division instead of the whole product.
Forgetting to change every sign when subtracting a polynomial.
Multiplying exponents instead of adding them when multiplying terms: $x^2 \cdot x^3 = x^5$ .
Writing the remainder without its divisor; $+7$ alone is not the same as $+\frac{7}{x - 2}$ .

Tips

Arrange both polynomials in descending powers before any operation.
Check a division by multiplying the quotient by the divisor and adding the remainder.
Use the remainder theorem to test a possible root quickly: if $P(c) = 0$ , then $x - c$ is a factor.
Sketch the graph with the graphing calculator to see how many real roots to expect.
Use the factoring calculator when you want the polynomial written as a product.

Frequently asked questions

What operations does the polynomial calculator support?