Polynomial Calculator
Choose expand, factor or solve, then enter a polynomial to see the result.
Use ^ for powers. For Solve, enter the polynomial that equals zero.
A polynomial calculator can expand a product into a sum of terms, factor a polynomial into a product of simpler polynomials, or solve for its roots (the values that make it zero). For example, (x + 2)(x + 3) expands to x² + 5x + 6, which factors back and has roots at x = −2 and x = −3.
How to Use This Calculator
Pick what you want to do. Expand multiplies out a product of brackets into a sum of terms. Factor does the reverse, writing a polynomial as a product of simpler factors. Solve finds the roots — the values of the variable that make the polynomial equal zero.
Enter the polynomial using ^ for powers, and the calculator returns the result for the chosen mode.
How the Calculation Works
Expanding applies the distributive law repeatedly until no brackets remain and like terms are combined. Factoring searches for common factors and known patterns to write the polynomial as a product, which is the inverse of expanding.
Solving sets the polynomial equal to zero and finds the values that satisfy it. For a factored polynomial, each factor set to zero gives a root, which is why factoring and solving are closely linked.
expand and factor are inverse operations; roots solve p(x) = 0
Worked Example
Factoring x² + 5x + 6 gives (x + 2)(x + 3), because those two numbers multiply to 6 and add to 5. Expanding (x + 2)(x + 3) returns x² + 5x + 6, confirming the two are inverses.
Solving x² + 5x + 6 = 0 gives the roots x = −2 and x = −3 — the values that make each factor zero.
Assumptions and Limitations
- Works with single-variable polynomials for factoring and solving
- Not every polynomial factors neatly over the integers
- Solve returns real and, where applicable, exact roots
- Results may be ordered differently from a textbook's convention
Who Uses a Polynomial Calculator
Algebra students use this to expand, factor, or solve a polynomial expression. It supports homework across a range of topics — from multiplying binomials to finding roots — and shows the result clearly. Enter a polynomial and choose what you need done.
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Open calculator →Frequently Asked Questions
What is the difference between expanding and factoring?
Expanding multiplies brackets out into a sum of terms; factoring writes a polynomial as a product of simpler factors. They are inverse operations — factoring then expanding returns the original.
How do you find the roots of a polynomial?
Set it equal to zero and solve. For a factored polynomial, each factor equal to zero gives a root. x² + 5x + 6 = 0 factors to (x+2)(x+3), giving roots −2 and −3.
Does every polynomial factor?
Over the integers, no. Some polynomials are irreducible or have irrational or complex roots. The calculator factors when it can and otherwise returns the polynomial or its exact roots.
What does 'solve' return for this tool?
It returns the roots — the variable values that make the polynomial equal zero. Enter the polynomial that equals zero, and it lists the solutions.
Can it handle high-degree polynomials?
It handles common cases well. Very high-degree polynomials may not factor over the integers, in which case it returns the exact or numeric roots where possible.
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