Polynomial Long Division Calculator
Divide two polynomials step-by-step, just like numerical long division. Find the quotient and remainder with a detailed explanation of each step.
Divide Polynomials
Enter the dividend and divisor polynomials using ^ for exponents (e.g., 2x^3 - 3x^2 + 0x - 5). Click Calculate to see the quotient and remainder.
0x) to maintain alignment if needed, though the calculator handles missing terms automatically.
Polynomial Long Division – Complete Explanation
Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. It generalizes the arithmetic long division process and is fundamental in algebra for simplifying expressions, finding asymptotes, and factoring polynomials.
Division Algorithm for Polynomials
Given polynomials P(x) and D(x) ≠ 0, there exist unique polynomials Q(x) and R(x) such that: P(x) = D(x) · Q(x) + R(x) where the degree of R(x) is less than the degree of D(x).
Q(x) is the quotient, R(x) is the remainder.
1. Steps for Polynomial Long Division
- Arrange terms in descending order of powers. Include missing terms with coefficient 0.
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from step 2.
- Subtract this product from the dividend.
- Bring down the next term if any, and repeat until the degree of the remainder is less than the degree of the divisor.
2. Step-by-Step Examples
Example 1: Basic Division with No Remainder
Divide: (2x³ - 3x² + 0x - 5) ÷ (x - 2) Step 1: Divide leading terms: 2x³ ÷ x = 2x² Step 2: Multiply: 2x²(x - 2) = 2x³ - 4x² Step 3: Subtract: (2x³ - 3x²) - (2x³ - 4x²) = x² Step 4: Bring down 0x → x² + 0x Step 5: Divide: x² ÷ x = x Step 6: Multiply: x(x - 2) = x² - 2x Step 7: Subtract: (x² + 0x) - (x² - 2x) = 2x Step 8: Bring down -5 → 2x - 5 Step 9: Divide: 2x ÷ x = 2 Step 10: Multiply: 2(x - 2) = 2x - 4 Step 11: Subtract: (2x - 5) - (2x - 4) = -1 Quotient: 2x² + x + 2 Remainder: -1
Example 2: Division Resulting in a Fraction
Divide: (4x³ - 5x + 2) ÷ (2x + 1) Step 1: 4x³ ÷ 2x = 2x² Step 2: 2x²(2x + 1) = 4x³ + 2x² Step 3: (4x³ + 0x²) - (4x³ + 2x²) = -2x² Step 4: Bring down -5x → -2x² - 5x Step 5: -2x² ÷ 2x = -x Step 6: -x(2x + 1) = -2x² - x Step 7: (-2x² - 5x) - (-2x² - x) = -4x Step 8: Bring down +2 → -4x + 2 Step 9: -4x ÷ 2x = -2 Step 10: -2(2x + 1) = -4x - 2 Step 11: (-4x + 2) - (-4x - 2) = 4 Quotient: 2x² - x - 2 Remainder: 4
3. More Examples
Example 1: (x² - 5x + 6) ÷ (x - 3)
Quotient: x - 2, Remainder: 0 | Factorization: (x - 3)(x - 2)
Example 2: (6x⁴ + 5x³ + 3x - 1) ÷ (3x² - 2x + 1)
Quotient: 2x² + 3x + 3, Remainder: 8x - 4
Example 3: (x³ - 8) ÷ (x - 2)
Quotient: x² + 2x + 4, Remainder: 0 (Difference of cubes)
4. Synthetic Division (Shortcut)
When dividing by a linear factor of the form (x - c), you can use synthetic division, which is much faster. It only uses the coefficients, not the variables.
Example: (3x³ - 4x² + 2x - 1) ÷ (x - 2)
Coefficients: 3, -4, 2, -1 | c = 2
Bring down 3. Multiply 3·2=6, add to -4 → 2. Multiply 2·2=4, add to 2 → 6. Multiply 6·2=12, add to -1 → 11.
Quotient: 3x² + 2x + 6, Remainder: 11
5. The Remainder Theorem
A direct consequence of the division algorithm: When a polynomial P(x) is divided by (x - c), the remainder is P(c).
Remainder Theorem
If P(x) = (x - c)·Q(x) + R, then P(c) = R
This provides a quick way to evaluate polynomials and is the basis for the Factor Theorem.
6. The Factor Theorem
(x - c) is a factor of P(x) if and only if P(c) = 0 (remainder is zero). This is a powerful tool for finding roots and factoring polynomials.
Example: Is (x - 3) a factor of x³ - 4x² + x + 6?
P(3) = 27 - 36 + 3 + 6 = 0 → Yes! Dividing gives (x - 3)(x² - x - 2) = (x - 3)(x - 2)(x + 1).
7. Common Mistakes to Avoid
- Missing terms: Forgetting to include 0x² or 0x as placeholders disrupts alignment.
- Sign errors during subtraction: Remember to subtract the entire product (change all signs).
- Stopping too early: Continue until the degree of the remainder is strictly less than the degree of the divisor.
- Incorrect leading term division: Always divide the highest-degree term of the current polynomial by the highest-degree term of the divisor.
- Miswriting the final answer: If remainder is not zero, express as Quotient + Remainder/Divisor.
8. Writing the Final Answer
If the remainder is not zero, the result of P(x) ÷ D(x) is written as:
P(x)/D(x) = Q(x) + R(x)/D(x)
Example: (2x³ - 3x² - 5) ÷ (x - 2) = 2x² + x + 2 - 1/(x - 2)
9. Real-World Applications
- Calculus: Integration by partial fractions requires polynomial division when the numerator's degree ≥ denominator's degree.
- Engineering: Simplifying transfer functions in control systems.
- Computer Graphics: Spline and curve calculations often involve polynomial division.
- Economics: Approximating cost and revenue functions.
- Cryptography: Polynomial arithmetic in finite fields.
10. Practice Problems with Solutions
Problem 1: Divide (x² - 9) ÷ (x + 3)
Solution: Quotient: x - 3, Remainder: 0
Problem 2: Divide (2x³ + 3x² - 1) ÷ (x² + 1)
Solution: Quotient: 2x + 3, Remainder: -2x - 4
Problem 3: Divide (4x⁴ - 2x² + x - 3) ÷ (2x - 1)
Solution: Quotient: 2x³ + x² - 0.5x + 0.25, Remainder: -2.75 (or use fractions)
11. Tips for Mastering Polynomial Division
- Always arrange both polynomials in descending order first.
- Use graph paper or careful alignment for manual calculations.
- Practice synthetic division for linear divisors to save time on tests.
- Check your answer by multiplying: Divisor × Quotient + Remainder should equal the Dividend.
- Understand the connection to factoring and roots for deeper algebraic insight.
12. Final Thoughts
Polynomial long division is a foundational algebraic skill that bridges arithmetic and advanced mathematics. While calculators can perform the operation instantly, understanding the process is crucial for grasping the structure of polynomials and their behavior.
Use this calculator to verify your manual work and explore patterns, but prioritize learning the step-by-step method to build lasting mathematical confidence.