Polynomial Division Calculator

Polynomial long division is analogous to numerical long division. You repeatedly divide the leading term of the current remainder by the leading term of the divisor, multiply, subtract, and bring down the next term.

Use synthetic division when dividing by a linear factor (x - r). It is faster and involves only the coefficients, avoiding writing out variables. Not suitable for higher-degree divisors.

A zero remainder means the divisor is a factor of the dividend. By the Factor Theorem, if P(r) = 0, then (x - r) is a factor of P(x).

Insert terms with coefficient 0 for any missing powers. For example, x³ + 1 should be written as x³ + 0x² + 0x + 1 to maintain proper alignment.

If the divisor has higher degree than the dividend, the quotient is 0 and the remainder is the dividend itself. The division algorithm still applies.

The Remainder Theorem states that when dividing P(x) by (x - r), the remainder equals P(r). This provides a quick way to evaluate polynomials.

Multiply the quotient by the divisor and add the remainder. The result should equal the original dividend: D(x) × Q(x) + R(x) = P(x).

Applications include: finding polynomial roots, simplifying rational expressions, partial fraction decomposition, checking factors, and analyzing polynomial behavior.

Long division works for any divisor degree and shows full algebraic manipulation. Synthetic division is a shortcut for linear divisors (x - r) that uses only coefficients, making it faster but more limited.

Step 1: Divide the leading terms. Step 2: Multiply the divisor by the result. Step 3: Subtract from the dividend. Step 4: Bring down the next term. Step 5: Repeat until the remainder degree is less than the divisor degree.

The Factor Theorem states that (x - r) is a factor of polynomial P(x) if and only if P(r) = 0. Polynomial division with remainder 0 confirms this.

No. Synthetic division only works for linear divisors like (x - r). For quadratic divisors, you must use polynomial long division or factor the divisor first.

A constant remainder means the divisor does not divide evenly. The result is written as Quotient + Remainder/Divisor. For example: 2x + 3 + 7/(x - 1).

Write the root r on the left (for divisor x - r). Write the coefficients of the dividend in order. Bring down the first coefficient, multiply by r, add to the next coefficient, and repeat.

The quotient is the result of dividing out the divisor. If the divisor is (x - r), the quotient gives the remaining factor after removing (x - r) from the polynomial.

The algorithm is identical: divide, multiply, subtract, bring down. The key difference is that polynomial division works with variable terms and powers rather than single digits.

Polynomial division is typically taught in Algebra 2 or Precalculus, usually in grades 9-11. It is essential for understanding calculus concepts like partial fractions.

Yes. Dividing a polynomial by itself gives quotient 1 and remainder 0, since any nonzero polynomial is a factor of itself.

The quotient is a constant (the ratio of leading coefficients). For example, (3x² + 2x) ÷ (6x² - x) = 1/2 + remainder terms.

By the Factor Theorem, if division by (x - r) gives remainder 0, then r is a root. This helps find rational roots and factor polynomials completely.

Before decomposing a rational expression, you must check if the numerator degree is less than the denominator degree. If not, polynomial long division is required first.

The Polynomial Division Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Polynomial Division Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.