Polynomial Division Calculator

Understanding Polynomial Division

Polynomial division is an algebraic algorithm for dividing one polynomial by another of equal or lower degree. Just like integer arithmetic where dividing two numbers yields a quotient and a remainder, dividing a dividend polynomial \(P(x)\) by a divisor polynomial \(D(x)\) yields a unique quotient \(Q(x)\) and remainder \(R(x)\).

The division algorithm for polynomials states that for any polynomial \(P(x)\) and non-zero divisor \(D(x)\), there exist unique polynomials \(Q(x)\) and \(R(x)\) such that:

$$P(x) = D(x) \cdot Q(x) + R(x)$$

where either \(R(x) = 0\) or the degree of \(R(x)\) is strictly less than the degree of \(D(x)\). In rational form, this is expressed as:

$$\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$$

To perform other algebraic operations on polynomials, you can use our Add and Subtract Polynomials Calculator and Multiplying Polynomials Calculator.

Polynomial Long Division Algorithm

Polynomial long division mirrors numerical long division. The systematic steps are:

1. Arrange terms: Write both the dividend and divisor in descending order of exponents, inserting zeros for any missing intermediate terms.
2. Divide leading terms: Divide the leading term of the dividend (or current remainder) by the leading term of the divisor to obtain the next term of the quotient.
3. Multiply: Multiply the entire divisor by the new quotient term.
4. Subtract: Subtract that product from the current remainder polynomial.
5. Repeat: Bring down the next term and repeat until the degree of the remaining expression is strictly less than the degree of the divisor.

For numerical arithmetic parallels, review our Long Division Calculator.

Synthetic Division Shortcut

Synthetic division is a streamlined shortcut for dividing polynomials when the divisor is a linear monic binomial of the form \(x - c\). Instead of carrying variable exponents through repeated subtractions, synthetic division works exclusively with coefficients:

• Write the root \(c\) to the left.
• List all coefficients of the dividend polynomial in order (including zeros for omitted powers).
• Bring down the first coefficient.
• Multiply the result by \(c\), place it under the next coefficient, and add the column.
• Repeat across all columns. The bottom row delivers the quotient coefficients, while the final number is the remainder.

The Remainder Theorem and Factor Theorem

Polynomial division has profound consequences in algebra through two celebrated theorems:

The Remainder Theorem: When a polynomial \(P(x)\) is divided by a linear factor \((x - c)\), the remainder is simply the value of the polynomial evaluated at \(c\), so \(R = P(c)\).

The Factor Theorem: A linear binomial \((x - c)\) is a factor of \(P(x)\) if and only if the remainder is zero (\(P(c) = 0\)). When the remainder is zero, the division is exact, confirming \(c\) is a real root of the polynomial.