Polynomial Division Calculator – Guide & Formulas
Divide polynomials using long division or synthetic division. Enter dividend and divisor to see quotient, remainder, and step-by-step division work for any polynomial.
Try the free calculator
Put these formulas into practice with our instant, step-by-step Polynomial Division Calculator.
Our free **Polynomial Division Calculator** performs both **long division** and **synthetic division** for any two polynomials. Enter a **dividend** (like 2x³ + 3x² - x + 5) and a **divisor** (like x + 1) to instantly see the **quotient** and **remainder** with detailed step-by-step work. Synthetic division is supported for **linear divisors** and provides a faster shortcut. Whether you are finding roots via the **Factor Theorem**, simplifying **rational expressions**, or preparing for **partial fraction decomposition**, this calculator handles all polynomial degrees with precision.
Key Takeaway
Use the free Polynomial Division Calculator to divide polynomials using long division or synthetic division. enter dividend and divisor to see quotient, remainder, and step-by-step division work for any polynomial. Get instant results with step-by-step explanations.
How to Use the Polynomial Division Calculator
- Enter the dividend polynomial (the polynomial being divided).
- Enter the divisor polynomial (the polynomial dividing by).
- Choose between long division and synthetic division (synthetic requires linear divisor).
- Review the quotient, remainder, and step-by-step division process.
- Verify by multiplying the quotient by the divisor and adding the remainder.
The Formula
Variable Definitions
- P(x): The dividend (polynomial being divided)
- D(x): The divisor (polynomial dividing by)
- Q(x): The quotient (result of division)
- R(x): The remainder (what is left over)
- Synthetic division: A shortcut method when dividing by a linear factor (x - r)
Dividing (2x³ + 3x² - x + 5) by (x + 1)
Perform polynomial long division.
- Step 1: Divide first terms: 2x³ ÷ x = 2x². Multiply: 2x²(x + 1) = 2x³ + 2x².
- Step 2: Subtract: (2x³ + 3x²) - (2x³ + 2x²) = x². Bring down -x.
- Step 3: Divide: x² ÷ x = x. Multiply: x(x + 1) = x² + x. Subtract: (x² - x) - (x² + x) = -2x.
- Step 4: Bring down +5. Divide: -2x ÷ x = -2. Multiply: -2(x + 1) = -2x - 2. Subtract: (-2x + 5) - (-2x - 2) = 7.
- Step 5: Quotient: 2x² + x - 2. Remainder: 7. Result: 2x² + x - 2 + 7/(x + 1).
Financial Advisory Notice
This calculator is for educational purposes only. Always verify results independently for academic or professional applications.
Frequently Asked Questions
What is polynomial long division?
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.
When should I use synthetic division?
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.
What does a zero remainder mean?
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).
How do I handle missing terms?
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.
Can I divide by a polynomial of higher degree?
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.
What is the Remainder Theorem?
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.
How do I verify my division?
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).
What are applications of polynomial division?
Applications include: finding polynomial roots, simplifying rational expressions, partial fraction decomposition, checking factors, and analyzing polynomial behavior.
What is the difference between long division and synthetic division?
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.
How do I divide polynomials step by step?
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.
What is the Factor Theorem?
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.
Can I use synthetic division for x² - 4?
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.
What if the remainder is a constant?
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).
How do I set up synthetic division?
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.
What does the quotient represent?
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.
Is polynomial division the same as long division of numbers?
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.
What grade level covers polynomial division?
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.
Can I divide a polynomial by itself?
Yes. Dividing a polynomial by itself gives quotient 1 and remainder 0, since any nonzero polynomial is a factor of itself.
What if both dividend and divisor are the same degree?
The quotient is a constant (the ratio of leading coefficients). For example, (3x² + 2x) ÷ (6x² - x) = 1/2 + remainder terms.
How does polynomial division help solve equations?
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.
What is the connection to partial fraction decomposition?
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.