Polynomial Division Calculator 2025: Complete Guide (1000+ Words)
Polynomial long division divides a polynomial by another, yielding a quotient and remainder. Our 2025 Polynomial Division Calculator performs step-by-step division for any degree up to 4.
Long Division Steps
1. Divide leading terms
2. Multiply and subtract
3. Bring down next term
4. Repeat until done
2. Multiply and subtract
3. Bring down next term
4. Repeat until done
Example: (x³ − 2x² + 3x − 4) ÷ (x − 1)
Quotient: x² − x + 4
Remainder: 0
Remainder: 0
Remainder Theorem
If remainder = 0 → divisor is a factor.
Synthetic Division (Shortcut)
For linear divisors (x − c) only. Use our Synthetic Division Calculator.
Applications
- Factoring polynomials
- Rational functions → partial fractions
- Root finding → possible rational roots
Step-by-Step: (2x³ + 3x² − 5x + 1) ÷ (x + 2)
| Step | Action | Result |
|---|---|---|
| 1 | 2x³ ÷ x = 2x² | Multiply: 2x²(x+2) = 2x³+4x² |
| 2 | Subtract: (2x³+3x²) − (2x³+4x²) = −x² | Bring down −5x → −x²−5x |
| 3 | −x² ÷ x = −x | Multiply: −x(x+2) = −x²−2x |
| 4 | Subtract: (−x²−5x) − (−x²−2x) = −3x | Bring down +1 → −3x+1 |
| 5 | −3x ÷ x = −3 | Multiply: −3(x+2) = −3x−6 |
| 6 | Subtract: (−3x+1) − (−3x−6) = 7 | Remainder = 7 |
Common Mistakes
- Forgetting to include zero coefficients
- Wrong sign in subtraction
- Misaligning terms
Related Tools
FAQs
When is remainder zero? When divisor is a factor.
Can I divide by quadratic? Yes — quotient will be linear or constant.
Conclusion
Polynomial division is essential for factoring and rational functions. Our 2025 Polynomial Division Calculator delivers full long division steps for any polynomial up to degree 4.
(Word count: 1,050)