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ALGEBRA / POLYNOMIALS

Polynomial Operations and Factoring Calculator with Division, Derivatives and Worked Steps

Updated By Unit Conversion Plus

Combine, divide, differentiate or factor polynomials. Enter coefficients from the highest power down, including zero for every missing term.

Enter the coefficients

Example: 1, −5, 6 represents x² − 5x + 6.

Polynomial arithmetic and factoring

Addition and subtraction align coefficients of equal powers. Multiplication distributes every term in one polynomial across every term in the other. Polynomial long division returns a quotient and a remainder, just as integer division does.

How the factoring check works

The calculator applies the rational root theorem when the coefficients are integers, tests possible rational zeros and uses synthetic division to remove each confirmed linear factor. Any factor that does not split into rational linear factors is left as a remaining polynomial.

Coefficient order matters

Enter coefficients in descending powers. For x³ − 4x + 3, enter 1, 0, −4, 3. The zero preserves the missing x² term and prevents every following coefficient from shifting to the wrong power.

Frequently asked questions

Can every polynomial be factored into integer factors?

No. Some polynomials are irreducible over the integers, while others require irrational or complex roots.

What does the remainder mean?

If P(x) is divided by Q(x), then P(x) = Q(x) times the quotient plus the remainder. The remainder has lower degree than Q(x).