Just as we can divide numbers, we can also divide polynomials. Polynomial division is a fundamental technique in algebra used to simplify complex rational expressions and to find the roots of polynomials. The process is analogous to the long division of integers you learned in elementary school, but applied to algebraic expressions. Our Polynomial Division Calculator helps students and professionals perform this operation quickly, providing the quotient and remainder for any two polynomials and making it easier to check work and understand the results of this important process.
How to Use the Polynomial Division Calculator
Dividing two polynomials is simple with our calculator:
- Enter the Dividend: In the first field, input the coefficients of the polynomial you are dividing *into*. Start with the coefficient of the highest degree term and separate each coefficient with a comma or space. Use '0' for any missing terms.
- Enter the Divisor: In the second field, input the coefficients of the polynomial you are dividing *by*.
- Calculate: Click the "Divide" button.
- View the Result: The calculator will display the resulting quotient and remainder polynomials.
The Process of Polynomial Long Division
Polynomial long division follows a systematic, repeating cycle of steps: Divide, Multiply, Subtract, Bring Down.
Let's divide (x³ - 3x² + 4x - 7) by (x - 1).
- Divide: Divide the leading term of the dividend (x³) by the leading term of the divisor (x). The result is x². This is the first term of your quotient.
- Multiply: Multiply your entire divisor (x - 1) by the term you just found (x²). The result is x³ - x².
- Subtract: Subtract this result from the original dividend. (x³ - 3x²) - (x³ - x²) = -2x².
- Bring Down: Bring down the next term from the dividend (+4x). Your new polynomial to work with is now -2x² + 4x.
- Repeat: Repeat the entire process. Divide the new leading term (-2x²) by the divisor's leading term (x) to get -2x. Multiply (x - 1) by -2x to get -2x² + 2x. Subtract this to get 2x. Bring down the -7. Repeat one last time to get the final remainder.
This systematic process continues until the degree of the remaining polynomial is less than the degree of the divisor. That final polynomial is your remainder.
The Result of Division
The result of any polynomial division can be expressed in the form:
Dividend / Divisor = Quotient + (Remainder / Divisor)
Our calculator provides you with the Quotient and Remainder separately.
Synthetic Division: A Powerful Shortcut
When your divisor is a simple linear binomial of the form (x - c), you can use a much faster and less cumbersome method called synthetic division. It is a shorthand way of performing polynomial long division that doesn't require you to write out the variables. Our Synthetic Division Calculator is designed for this specific, common case.
Applications of Polynomial Division
- Finding Roots/Zeros: The Remainder Theorem states that if you divide a polynomial f(x) by (x - c), the remainder will be equal to f(c). A direct consequence is the Factor Theorem: if the remainder is 0, then (x - c) is a factor of the polynomial, and 'c' is a root. This is the primary method used to find all roots of a high-degree polynomial after one root is known.
- Simplifying Rational Expressions: It is used to simplify "improper" rational functions (where the degree of the numerator is greater than or equal to the degree of the denominator) as a required first step before performing other operations, like partial fraction decomposition in calculus.
- Finding Asymptotes: In pre-calculus, polynomial division is used to find the slant (or oblique) asymptotes of the graphs of rational functions.
Frequently Asked Questions
What do I do if my polynomial has a missing term?
It is crucial that you account for any missing terms by using a zero as a placeholder coefficient. For example, if you are dividing x³ + 2x - 5, you must enter its coefficients as "1, 0, 2, -5" to represent the missing x² term. Failing to include the zero will lead to an incorrect result.
What does it mean if the remainder is zero?
A remainder of zero means that the divisor is a factor of the dividend. The dividend can be written as the product of the divisor and the quotient with nothing left over. This also means that the roots of the divisor are also roots of the dividend.
Can I use this for a divisor that isn't linear?
Yes. This calculator performs full polynomial long division and can handle divisors of any degree, including quadratic or cubic divisors. It is not limited to linear divisors like a synthetic division calculator is.