Polynomial Division Calculator
Free Polynomial division Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Polynomial Division Calculator
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Formula: P(x) = Q(x) * D(x) + R(x)
Worked example — (x^3 - 7x + 6) / (x - 1) = x^2 + x - 6 | Remainder: 0
Formula
P(x) = Q(x) * D(x) + R(x)
The division algorithm states that for polynomials P(x) (dividend) and D(x) (divisor, nonzero), there exist unique polynomials Q(x) (quotient) and R(x) (remainder) such that P(x) = Q(x) * D(x) + R(x), where deg(R) < deg(D) or R = 0.
Worked Examples
Example 1: Long Division of Cubic by Linear
Problem:Divide x^3 - 7x + 6 by (x - 1) using polynomial long division.
Solution:Dividend coefficients: 1, 0, -7, 6 (note the 0 for missing x^2 term) Divisor coefficients: 1, -1 Step 1: x^3 / x = x^2. Multiply (x-1) by x^2: x^3 - x^2. Subtract. Step 2: x^2 / x = x. Multiply (x-1) by x: x^2 - x. Subtract. Step 3: -6x / x = -6. Multiply (x-1) by -6: -6x + 6. Subtract. Quotient: x^2 + x - 6 Remainder: 0 Verify: P(1) = 1 - 7 + 6 = 0. Factor Theorem confirmed! Further factoring: x^2 + x - 6 = (x + 3)(x - 2)
Result:(x^3 - 7x + 6) / (x - 1) = x^2 + x - 6 | Remainder: 0
Example 2: Synthetic Division with Remainder
Problem:Use synthetic division to divide 2x^3 + x^2 - 13x + 6 by (x - 2).
Solution:Root: c = 2 Coefficients: 2, 1, -13, 6 Bring down 2. 2 * 2 = 4. Add to 1: 5. 2 * 5 = 10. Add to -13: -3. 2 * (-3) = -6. Add to 6: 0. Result row: 2, 5, -3, 0 Quotient: 2x^2 + 5x - 3 Remainder: 0 Since remainder is 0, (x - 2) is a factor. 2x^2 + 5x - 3 = (2x - 1)(x + 3)
Result:(2x^3 + x^2 - 13x + 6) / (x - 2) = 2x^2 + 5x - 3 | Remainder: 0
Frequently Asked Questions
What is polynomial long division and when should you use it?
Polynomial long division is an algorithm for dividing one polynomial by another, analogous to long division of numbers. It produces a quotient polynomial and a remainder polynomial such that dividend = (quotient)(divisor) + remainder. You should use polynomial long division whenever the divisor has degree 2 or higher, or when you need to see the step-by-step process. The algorithm works by repeatedly dividing the leading term of the current dividend by the leading term of the divisor, multiplying the result through, and subtracting. This process continues until the remaining polynomial has a degree less than the divisor. Polynomial division is fundamental for factoring, finding oblique asymptotes of rational functions, and simplifying complex rational expressions.
What is synthetic division and how does it differ from long division?
Synthetic division is a streamlined shortcut for dividing a polynomial by a linear divisor of the form (x - c). It uses only the coefficients and is significantly faster than long division, but it only works when the divisor is linear with a leading coefficient of 1. The process writes the root c on the left, lists the dividend coefficients across the top, then uses a bring-down-multiply-add pattern. The last number in the result is the remainder, and the other numbers are the quotient coefficients. For example, dividing x^3 - 6x^2 + 11x - 6 by (x - 2): write 2 on the left and coefficients 1, -6, 11, -6 across the top. The result gives quotient x^2 - 4x + 3 with remainder 0, confirming (x - 2) is a factor.
What is the Remainder Theorem and how does it relate to polynomial division?
The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder equals P(c). This means you can find the remainder without performing the entire division: simply evaluate the polynomial at x = c. For example, dividing P(x) = x^3 + 2x - 5 by (x - 3): P(3) = 27 + 6 - 5 = 28, so the remainder is 28. This theorem has a powerful corollary called the Factor Theorem: (x - c) is a factor of P(x) if and only if P(c) = 0. These theorems together provide a systematic way to test potential factors of a polynomial and are essential in finding roots and factoring higher-degree polynomials.
How do you find oblique (slant) asymptotes using polynomial division?
When the degree of the numerator of a rational function is exactly one more than the degree of the denominator, the function has an oblique or slant asymptote. To find it, perform polynomial long division. The quotient (ignoring the remainder) gives the equation of the oblique asymptote. For example, for f(x) = (x^2 + 3x + 5)/(x + 1), dividing gives x + 2 with remainder 3. The oblique asymptote is y = x + 2. As x approaches infinity, the remainder term 3/(x+1) approaches zero, so the function approaches the line y = x + 2. This technique is essential for sketching the behavior of rational functions and is commonly tested in precalculus and calculus courses.
What happens when the divisor does not divide evenly?
When the divisor does not divide evenly into the dividend, you get a nonzero remainder. The complete result is expressed as: dividend/divisor = quotient + remainder/divisor. For example, (x^3 + 2x + 1)/(x^2 + 1) = x + (x + 1)/(x^2 + 1). The degree of the remainder must always be strictly less than the degree of the divisor. This representation is analogous to how 17 divided by 5 equals 3 remainder 2, written as 17/5 = 3 + 2/5. The division algorithm guarantees that the quotient and remainder are unique. In calculus, this mixed form is often easier to integrate because the quotient is a simple polynomial and the remainder fraction may yield to partial fractions or substitution.
How do you verify that polynomial division was done correctly?
The primary verification method is the division algorithm identity: dividend = (quotient)(divisor) + remainder. Multiply the quotient by the divisor, add the remainder, and confirm the result equals the original dividend. You can also use numerical substitution: choose a value for x, evaluate all four expressions (dividend, divisor, quotient, remainder), and verify that dividend = quotient times divisor plus remainder at that value. A third check is the Remainder Theorem: if dividing by (x - c), evaluate the dividend at x = c and confirm it equals the remainder. Additionally, check that the degree of the quotient equals (dividend degree minus divisor degree) and that the remainder degree is strictly less than the divisor degree.
How is polynomial division used in factoring polynomials?
Polynomial division is the primary tool for factoring polynomials of degree 3 and higher. The strategy is: first find one root c (using the Rational Root Theorem, graphing, or trial), then divide the polynomial by (x - c) to reduce the degree. If P(c) = 0 by the Factor Theorem, the division gives zero remainder and produces a polynomial one degree lower. Repeat this process on the quotient. For example, to factor x^3 - 6x^2 + 11x - 6: test x = 1, get P(1) = 0, so (x - 1) is a factor. Divide to get x^2 - 5x + 6. Factor this quadratic: (x - 2)(x - 3). Complete factorization: (x - 1)(x - 2)(x - 3). This divide-and-conquer approach systematically reduces any polynomial to its linear and irreducible quadratic factors.
What is the Rational Root Theorem and how does it connect to division?
The Rational Root Theorem states that any rational root p/q of a polynomial with integer coefficients must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem generates a finite list of candidates to test. For x^3 - 2x^2 - 5x + 6, the constant term is 6 (factors: 1, 2, 3, 6) and the leading coefficient is 1 (factor: 1), so candidates are plus or minus 1, 2, 3, 6. Test each using the Remainder Theorem: P(1) = 0, so (x - 1) is a factor. Then perform polynomial division to find the remaining factors. The Rational Root Theorem narrows the search from infinitely many possibilities to a manageable list, making it an indispensable first step in polynomial factoring.
Can you divide polynomials with missing terms or non-integer coefficients?
Yes, polynomial division works with any real or even complex coefficients, including fractions and decimals. When terms are missing (for example, x^3 + 5 has no x^2 or x terms), you must include zero coefficients as placeholders. Enter x^3 + 5 as the coefficient list 1, 0, 0, 5. Forgetting zero placeholders is the most common error in polynomial division because it misaligns the degree of each term. The algorithm works identically regardless of coefficient type. For non-integer results, the quotient and remainder may have fractional coefficients. For instance, dividing 2x^2 + 3x + 1 by 3x + 1 gives quotient (2/3)x + (7/9) with remainder 2/9.
What are the applications of polynomial division in advanced mathematics?
Polynomial division has far-reaching applications beyond basic algebra. In abstract algebra, it underpins the theory of polynomial rings and is used to construct finite fields essential to modern cryptography and error-correcting codes. In numerical analysis, polynomial division appears in deflation methods for finding all roots of a polynomial. In control theory, transfer function simplification relies on polynomial division. Computer algebra systems use it as a subroutine for GCD computation, resultants, and Groebner basis calculations. In coding theory, cyclic redundancy checks (CRC) used for data integrity are performed via polynomial division over binary fields. The Euclidean algorithm for polynomials, which finds the GCD of two polynomials, repeatedly applies polynomial division.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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