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Polynomial Calculator
Enter two polynomials P(x) and Q(x) and add, subtract, multiply, divide or evaluate them. Every result comes with the work, a property table (degree, leading coefficient, factored form, exact roots, end behavior) and an interactive graph.
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Examples:
Use x as the variable: + − * ^ and parentheses, e.g. 3x^4 - x + 2 or (x - 1)(x + 4). Fractions (x/3) and decimals (0.25x) are kept exact.
Result
Work
Properties
Graph
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How to use the Polynomial Calculator
- Enter P(x) and Q(x), for example x^3 - 2x^2 - 5x + 6 and x^2 + x - 2.
- Pick an operation: P + Q, P − Q, P × Q, P ÷ Q or Evaluate.
- Set a to the x-value you want to evaluate (optional for other operations).
- Read the result and the work, then check the property table.
- Explore the graph: drag to pan, scroll or use + and − to zoom.
Polynomial operations at a glance
| Operation | How it works | Degree of the result | Example |
|---|---|---|---|
| P + Q | Add coefficients of like powers | at most max(deg P, deg Q) | (x² + 3x) + (2x² − x + 4) = 3x² + 2x + 4 |
| P − Q | Change the signs of Q, then add | at most max(deg P, deg Q) | (x² + 3x) − (x² − x) = 4x |
| P × Q | Distribute every term, combine like terms | deg P + deg Q | (x + 2)(x − 3) = x² − x − 6 |
| P ÷ Q | Long division: P = Q·q + r | deg P − deg Q (quotient) | (x² − 1) ÷ (x − 1) = x + 1 |
| P(a) | Substitute a (Horner's method) | a number | P(x) = x² − 3, P(2) = 1 |
Notice that a sum can drop in degree when the leading terms cancel, as in the subtraction example, but a product of nonzero polynomials never does.
Reading the property table
- Factored form is written over the rational numbers. A factor such as x² + 1 or x² − 2 stays as a quadratic because it has no rational roots.
- Roots are listed with their multiplicity. A root of even multiplicity touches the x-axis without crossing it; an odd multiplicity crosses.
- Complex roots of a polynomial with real coefficients always come in conjugate pairs a ± bi, so a polynomial of odd degree has at least one real root.
- The number of roots, counted with multiplicity and including complex ones, equals the degree (the fundamental theorem of algebra).
- The derivative lowers the degree by one; its roots are where the graph can turn around.
Frequently asked questions
How do you add or subtract polynomials?
Combine like terms: add or subtract the coefficients of the same power of x. When subtracting, change the sign of every term of the second polynomial first. For example (3x² + 2x − 1) − (x² − 4x + 5) = 2x² + 6x − 6.
How do you multiply polynomials?
Multiply every term of the first polynomial by every term of the second (the distributive property) and then combine like terms. The degree of the product is the sum of the two degrees.
What is the degree and leading coefficient of a polynomial?
The degree is the highest power of x with a nonzero coefficient, and the leading coefficient is that coefficient. In −4x⁵ + x² − 7 the degree is 5 and the leading coefficient is −4. Together they decide the end behavior of the graph.
How are the roots found?
The calculator factors the polynomial over the rationals using the rational root theorem and then solves any quadratic factors exactly with the quadratic formula, so roots like (1 + √5)/2 or 2 ± 3i are shown exactly. Factors of degree 3 or more with no rational roots are solved numerically.
What does evaluating a polynomial mean?
Evaluating P at x = a means substituting a for x and simplifying, giving the number P(a). Horner's method does this efficiently: start with the leading coefficient, then repeatedly multiply by a and add the next coefficient.
What is end behavior?
End behavior describes where the graph goes as x → ±∞. For an even degree both ends point the same way (up if the leading coefficient is positive); for an odd degree they point opposite ways.
What happens when I divide polynomials?
Division gives a quotient and a remainder with P = Q·quotient + remainder, where the remainder has a lower degree than Q. If the remainder is 0, Q is a factor of P. For a full layout use the polynomial long division calculator.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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