1
-2
2
1
0
1
1
-2
2
1
0
1
The Polynomial Division Calculator performs long division of a cubic polynomial by a linear divisor, giving you the quotient and remainder instantly. Polynomial division is a core algebraic operation that extends the familiar concept of integer division into the realm of polynomial expressions, and it is essential for factoring, simplifying rational expressions, and solving higher-degree equations.
In standard arithmetic, dividing 17 by 5 gives a quotient of 3 and a remainder of 2, expressed as 17 = 5 × 3 + 2. Polynomial division works analogously: given a dividend polynomial P(x) and a divisor polynomial D(x), we find a quotient Q(x) and a remainder R(x) such that P(x) = D(x) × Q(x) + R(x), where the degree of R(x) is strictly less than the degree of D(x). This is known as the Division Algorithm for Polynomials, and it guarantees a unique quotient and remainder.
This calculator divides a cubic polynomial P(x) = ax³ + bx² + cx + d by a linear divisor D(x) = ex + f. The quotient will be a quadratic polynomial Q(x) = q₂x² + q₁x + q₀, and the remainder will be a constant. The division process mirrors long division: at each step, we divide the leading term of the current dividend by the leading term of the divisor, multiply the entire divisor by that result, and subtract to get a new (reduced) dividend.
One of the most important applications of polynomial division is factoring. If the remainder is zero, then the divisor D(x) divides P(x) evenly, meaning D(x) is a factor of P(x). Combined with the Remainder Theorem, which states that when P(x) is divided by (x − r) the remainder equals P(r), this gives a powerful method for testing potential roots. If P(r) = 0, then (x − r) is a factor, and the quotient Q(x) gives the remaining factor.
This connection between division and root-finding is central to the Rational Root Theorem, which lists all possible rational roots of a polynomial with integer coefficients. By systematically testing these candidates using polynomial division (or its streamlined cousin, synthetic division), you can fully factor many polynomials.
Polynomial division also plays a crucial role in partial fraction decomposition, a technique used extensively in calculus to break down complex rational expressions into simpler fractions that can be integrated individually. Before applying partial fractions, you must first use polynomial division to ensure the degree of the numerator is less than the degree of the denominator.
In abstract algebra, polynomial division over various coefficient fields forms the foundation of ring theory and ideal theory. The division algorithm ensures that the ring of polynomials over a field is a Euclidean domain, which implies unique factorization and enables the computation of greatest common divisors via the Euclidean algorithm for polynomials.
The calculator also provides a verification step by evaluating P(r) directly, where r is the root of the divisor (i.e., if the divisor is x − r, then r = −f/e). By the Remainder Theorem, this value should equal the computed remainder, serving as a built-in consistency check on the computation.
This calculator divides a cubic polynomial by a linear polynomial using the algebraic long division algorithm.
Setup: We divide P(x) = a₃x³ + a₂x² + a₁x + a₀ by D(x) = b₁x + b₀.
Step 1: Divide the leading term of P(x) by the leading term of D(x):
$$q_2 = \frac{a_3}{b_1}$$
Step 2: Multiply D(x) by q₂ and subtract from P(x). The new x² coefficient becomes:
$$a_2' = a_2 - q_2 \cdot b_0$$
Step 3: Divide the new leading term by b₁:
$$q_1 = \frac{a_2'}{b_1}$$
Step 4: Repeat to get the constant quotient term:
$$a_1' = a_1 - q_1 \cdot b_0, \quad q_0 = \frac{a_1'}{b_1}$$
Step 5: The remainder is:
$$R = a_0 - q_0 \cdot b_0$$
Result:
$$P(x) = D(x) \cdot (q_2 x^2 + q_1 x + q_0) + R$$
Verification via Remainder Theorem: If D(x) = b₁x + b₀, its root is r = −b₀/b₁. Then P(r) should equal R.
The quotient coefficients (q₂, q₁, q₀) define the quadratic polynomial Q(x) = q₂x² + q₁x + q₀ that results from the division. If you multiply Q(x) by the divisor D(x) and add the remainder, you recover the original polynomial P(x).
The remainder is a constant value. If it equals zero, the divisor divides the polynomial exactly, meaning the divisor is a factor. The "Is Factor" indicator shows 1 for yes and 0 for no.
The verification value P(r) computes the polynomial at the root of the divisor. By the Remainder Theorem, this must equal the remainder. If these values match, the computation is confirmed correct.
When the divisor's x coefficient (b₁) is zero, the divisor becomes a constant, and division by a constant is trivial (just divide each coefficient). The calculator handles this edge case by returning zeros to avoid division by zero errors.
Inputs
Results
Step 1: x³ ÷ x = x², so q₂ = 1. Multiply (x − 2) by x² = x³ − 2x². Subtract: (−4x²) − (−2x²) = −2x². Step 2: −2x² ÷ x = −2x, so q₁ = −2. Multiply (x − 2) by −2x = −2x² + 4x. Subtract: 6x − 4x = 2x. Step 3: 2x ÷ x = 2, so q₀ = 2. Multiply (x − 2) by 2 = 2x − 4. Subtract: −3 − (−4) = 1. Remainder = 1. Verification: P(2) = 8 − 16 + 12 − 3 = 1 ✓.
Inputs
Results
Dividing by (x + 2): q₂ = 2/1 = 2. New x² term: 3 − 2(2) = −1, so q₁ = −1. New x term: −1 − (−1)(2) = 1, so q₀ = 1. Remainder: −6 − 1(2) = −8. The divisor is not a factor since the remainder is −8. Check: P(−2) = 2(−8) + 3(4) + (−1)(−2) − 6 = −16 + 12 + 2 − 6 = −8 ✓.
If the divisor's x coefficient (b₁) is zero, the divisor becomes a constant (just b₀), and you would be dividing by a constant rather than a polynomial. In that case, simply dividing each coefficient of the dividend by the constant gives the result. The calculator returns zeros to avoid division by zero; for constant divisors, manually divide each coefficient by b₀ instead.
The Remainder Theorem states that when a polynomial P(x) is divided by a linear divisor (x − r), the remainder equals P(r). This means you can find the remainder without performing the full division—just evaluate the polynomial at x = r. This calculator uses this property as a verification: it computes both the algebraic remainder and P(r) to confirm they match.
This calculator is specifically designed for division by a linear polynomial (degree 1). For dividing by quadratics or higher-degree polynomials, you would need a more general polynomial long division tool. However, in practice, most textbook and applied problems involve division by linear factors, which is what this tool handles.
Both produce the same result. Long division mirrors the familiar format of numerical long division and works for any divisor. Synthetic division is a streamlined shortcut that only works when dividing by a linear divisor of the form (x − r) with a leading coefficient of 1. Synthetic division uses only the coefficients, making it faster and less error-prone for hand calculations.
In calculus, partial fraction decomposition requires that the degree of the numerator be less than the degree of the denominator. If it is not, you must first perform polynomial division to separate the "polynomial part" from the proper fraction. The polynomial part integrates directly using the power rule, and the proper fraction can then be decomposed into simpler fractions.
First, find one root r (using the Rational Root Theorem or by testing values). Then divide P(x) by (x − r) using this calculator. If the remainder is zero, the quotient Q(x) is a quadratic. Factor Q(x) using the quadratic formula or by inspection. This gives the complete factorization P(x) = (x − r)(q₂x² + q₁x + q₀).
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Polynomial Calculator
Polynomial Calculators
Synthetic Division Calculator
Polynomial Calculators
Factoring Calculator
Polynomial Calculators
Factoring Trinomials Calculator
Polynomial Calculators
Perfect Square Trinomial Calculator
Polynomial Calculators
Difference of Two Squares Calculator
Polynomial Calculators
