Polynomial Division Calculator – Divide Polynomials
Divide polynomials and get quotient and remainder
Table of Contents
How to Use
- Enter coefficients for the dividend polynomial (up to degree 3)
- Enter coefficients for the divisor polynomial (up to degree 2)
- Leave coefficients as 0 for missing terms
- Click calculate to see the quotient and remainder
- Result shows the division in polynomial form
What is Polynomial Division?
Polynomial division is the process of dividing one polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and possibly a remainder. It's similar to long division with numbers but works with algebraic expressions.
The division can be expressed as: Dividend = Divisor × Quotient + Remainder
Polynomial Long Division Method
To perform polynomial long division:
- Arrange both polynomials in descending order of degree
- Divide the leading term of the dividend by the leading term of the divisor
- Multiply the entire divisor by this result
- Subtract this product from the dividend
- Repeat with the new polynomial until the degree of the remainder is less than the degree of the divisor
Example
Divide x³ + 2x² - 5x + 6 by x + 2:
- x³ ÷ x = x² (first term of quotient)
- Multiply: (x + 2) × x² = x³ + 2x²
- Subtract: (x³ + 2x² - 5x + 6) - (x³ + 2x²) = -5x + 6
- Continue: -5x ÷ x = -5
- Multiply: (x + 2) × (-5) = -5x - 10
- Subtract: (-5x + 6) - (-5x - 10) = 16
- Result: Quotient = x² - 5, Remainder = 16
Applications of Polynomial Division
- Simplifying rational expressions
- Finding zeros and factors of polynomials
- Partial fraction decomposition in calculus
- Signal processing and filter design
- Solving polynomial equations
- Curve fitting and interpolation
- Control systems engineering
The Remainder Theorem
The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a). This provides a quick way to evaluate polynomials and check if (x - a) is a factor.
If the remainder is 0, then (x - a) is a factor of P(x), which is the Factor Theorem.
Frequently Asked Questions
- What is the difference between polynomial division and regular division?
- Polynomial division works with algebraic expressions containing variables, while regular division works with numbers. The process is similar to long division, but you divide terms with variables and combine like terms during subtraction.
- Can the remainder be zero?
- Yes, when the remainder is zero, it means the divisor is a factor of the dividend. The dividend can be expressed as the product of the divisor and quotient with no remainder.
- What if the dividend has a lower degree than the divisor?
- If the dividend has a lower degree than the divisor, the quotient is 0 and the remainder is the original dividend. You cannot divide a lower-degree polynomial by a higher-degree polynomial to get a non-zero quotient.
- How do I handle missing terms in a polynomial?
- For missing terms, use a coefficient of 0. For example, if you have x³ + 5 (missing x² and x terms), enter coefficients as: 1, 0, 0, 5.