This polynomial calculator adds, subtracts and multiplies two quadratic polynomials and shows the complete result term by term.

P(x) = a₁x² + b₁x + c₁

Q(x) = a₂x² + b₂x + c₂

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Like Terms Combine; Multiplication Convolves Every Term

Roots of a quadratic result belong on our Quadratic Formula Calculator. Part of our Math Calculators collection.

Worked Example (Verified)

  • (x + 2)(x − 3) using the defaults: x² − 3x + 2x − 6 = x² − x − 6
  • Addition example: (2x² + 3x − 1) + (x² − 2x + 4) = 3x² + x + 3
  • Subtraction is ordered: P − Q is generally not Q − P; only the constant and like-power coefficients change signs in Q.
  • Degree rule: degrees add under multiplication, so two genuine quadratics produce a quartic unless leading terms cancel in a special case.

Reading the Result

  • Coefficients are signed numbers; a displayed “− x” means coefficient −1, not a missing number.
  • Zero terms are omitted in standard form, except when the entire result is zero.
  • Factoring is the reverse journey: x² − x − 6 returns to (x + 2)(x − 3).
  • For cubics and higher, extend the same convolution pattern row by row.

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Frequently Asked Questions

How do you multiply polynomials?

Multiply every term in the first polynomial by every term in the second, then combine like powers. (x + 2)(x − 3) expands to x² − 3x + 2x − 6, which simplifies to x² − x − 6. This calculator performs that convolution for two quadratics.

What is the degree of the result?

For addition or subtraction, the degree is the highest surviving power. For multiplication, degrees add: quadratic × quadratic gives degree 4 when both leading coefficients are nonzero.

Can this calculator divide polynomials?

No — this page covers addition, subtraction and multiplication. Division by a linear factor x − c is handled by the Synthetic Division Calculator linked above.

Why do only like terms combine?

Terms such as 3x² and 2x represent different powers and cannot be merged into one coefficient without changing the function. They can sit in the same polynomial, but simplification only combines identical powers.

Does order matter?

Addition and multiplication are commutative: P + Q = Q + P and P × Q = Q × P. Subtraction is not; reversing it negates every term of the result.

Is this polynomial calculator free?

Yes — free, instant, no sign-up.