Polynomial Division Calculator
Perform polynomial long division with step-by-step solutions and remainder calculation
Polynomial Division Calculator
P(x) = x^3 - 5x + 6
Q(x) = x - 2
Division Results
Quotient:
Remainder:
Step-by-Step Long Division
Example Problem
Divide x³ - 5x + 6 by x - 2
Dividend: x³ - 5x + 6
Divisor: x - 2
Expected: x² + 2x - 1
Division Process
Divide leading term of dividend by leading term of divisor
Multiply entire divisor by the result
Subtract from dividend to get new dividend
Repeat until remainder degree < divisor degree
Division Tips
Always arrange polynomials in descending order of powers
Include zero coefficients for missing terms
Check your work: Q(x) * D(x) + R(x) = P(x)
Remainder degree must be less than divisor degree
Understanding Polynomial Division
What is Polynomial Division?
Polynomial division is a method for dividing one polynomial by another polynomial of lower or equal degree. It follows the same basic principles as arithmetic long division but works with algebraic expressions.
The Division Algorithm
For polynomials P(x) and Q(x) where Q(x) ≠ 0, there exist unique polynomials A(x) and R(x) such that:
P(x) = Q(x) * A(x) + R(x)
where degree of R(x) < degree of Q(x)
Key Terms
- •P(x): Dividend (polynomial being divided)
- •Q(x): Divisor (polynomial dividing into P(x))
- •A(x): Quotient (result of division)
- •R(x): Remainder (what's left over)
Long Division Steps
- 1
Setup: Arrange both polynomials in descending order of powers
- 2
Divide: Divide the leading term of dividend by leading term of divisor
- 3
Multiply: Multiply the entire divisor by the result from step 2
- 4
Subtract: Subtract the result from the current dividend
- 5
Repeat: Use the result as the new dividend and repeat until the remainder degree is less than the divisor degree
Applications
- • Finding polynomial roots and factors
- • Simplifying rational functions
- • Partial fraction decomposition
- • Solving polynomial equations
- • Mathematical proof techniques